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Dynamic Analysis of High-Speed Railway Bridges Effect of the Deck Cross Section Distortion
Mariana Vieira Coelho Antunes
Thesis to obtain the Master of Science Degree in
Civil Engineering
Supervisors: Francisco Baptista Esteves Virtuoso, PhD
Ricardo José de Figueiredo Mendes Vieira, PhD
Examination Committee
Chairperson: José Joaquim Costa Branco de Oliveira Pedro, PhD
Supervisor: Francisco Baptista Esteves Virtuoso, PhD
Members of the Committee: Luís Manuel Coelho Guerreiro, PhD
September 2017
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Abstract
The dynamic behaviour induced by the passage of high-speed trains plays an important role on the
design and performance of railway bridge decks.
The positioning of two opposite-way lanes eccentric to the cross-sectional shear centre implies the
existence of coupled bending and torsional vibrations. Furthermore, there are also vibration modes that
correspond to the transversal deformation of the cross section that may be relevant to the
determination of the dynamic response of the bridge.
This work aims to evaluate the dynamic response of railway bridge decks through finite element
models, due to the dynamic effects covered by the High-Speed Load Model (HSLM).The dynamic
behaviour of the deck cross section is considered through the shell and frame finite element models,
being evaluated the importance of several section deformation modes in the dynamic behaviour of the
bridge, and allowing the identification of resonance phenomena due to passage of trains defined in
HSLM and the evaluation of the influence of non-conventional deformation modes (e.g. distortion,
warping) on the dynamic behaviour of bridges.
In the end of the document are present examples illustrating the assessment of the resonance
phenomena in railway bridge decks for trains defined by the HSLM and allowing the identification of
the local section deformation modes with higher participation in the dynamic behaviour of the bridges.
Key Words:
High-speed rail bridges
Moving Loads
Dynamic analysis
Distortion
Acceleration
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Resumo
O comportamento dinâmico devido à passagem de comboios de alta velocidade desempenha um papel
importante na concepção e dimensionamento de tabuleiros de pontes ferroviárias.
A existência de duas vias de sentido oposto, excêntricas em relação ao centro de corte da secção
transversal leva ao aparecimento de vibrações de flexão e torção acopladas. Para além disso existem
também modos de vibração que correspondem à deformação transversal da secção transversal, os
quais podem ser relevantes para determinar a resposta dinâmica da ponte.
Neste trabalho avalia-se a resposta dinâmica de pontes ferroviárias através de modelos de elementos
finitos, devido aos efeitos dinâmicos produzidos pelo modelo de cargas móveis de alta velocidade -
HSLM. O comportamento dinâmico da seção transversal é considerado através de elementos finitos de
casca e de barra, tornando possível avaliar a importância dos vários modos de deformação da seção no
comportamento dinâmico da ponte, permitindo identificar fenómenos de ressonância devidos à
passagem de comboios HSLM e avaliar a influência de modos de deformação não convencionais (p.
ex. distorção, empenamento) no comportamento dinâmico das pontes.
No final do documento são apresentados exemplos ilustrativos da avaliação dos fenômenos de
ressonância em pontes ferroviárias para comboios definidos pelo HSLM e permitindo a identificação
dos modos de deformação da seção transversal com maior participação no comportamento dinâmico
das pontes.
Palavras Chave:
Pontes ferroviárias de alta velocidade
Cargas Móveis
Análise dinâmica
Distorção
Aceleração
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Acknowledgements
I would like to express my gratitude to everyone who somehow have contributed to the development
of the present work. However, I would also like to address a special thank to:
Professor Francisco Virtuoso, my supervisor, for his guidance, encouraging and scientific contribute.
His efforts were absolutely essential to take this dissertation this far.
Professor Ricardo Vieira, my co-supervisor, for his advices and comments, which significantly shaped
the final outcome of this thesis.
My close friends, with whom I have shared these last fantastic years, for having been a source of
motivation and for being pleasure to work with them.
João Cardoso, for his enormous contribution and for having given me a constant support throughout
this period that was fundamental to get this far.
Marta Antunes, my sister, for her help and also for supported me, throughout my whole academic
course, to achieve so many things.
My parents and grandparents, who have given me the opportunity to get here and who have always
supported me unconditionally.
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Contents
1. Introduction ..................................................................................................................................... 1
1.1 The Aim of the Work .............................................................................................................. 2
1.2 Literature review ..................................................................................................................... 3
1.3 Layout of the work .................................................................................................................. 6
2. Dynamic Analysis ........................................................................................................................... 9
2.1 Vibration Frequencies and Mode Shapes .................................................................................... 10
2.1.1 Modal Coordinates ............................................................................................................... 12
2.2 Viscous Damping ........................................................................................................................ 13
2.3 Forced Vibration.......................................................................................................................... 14
2.3.1 Modal Superposition ............................................................................................................ 15
2.3.2 Numeric Integration ............................................................................................................. 16
3. Dynamic Response of Beams under Moving Loads ..................................................................... 19
3.1 Simple Supported Beam .............................................................................................................. 20
3.1.1 Analytical Solution ............................................................................................................... 22
3.1.2 Numerical Solution ............................................................................................................... 28
3.2 Continuous Beam ........................................................................................................................ 33
3.2.1 Continuous beam under concentrated load ........................................................................... 34
3.2.2 Continuous beam applied for a set of concentrated loads .................................................... 37
4. Deck Modelling - warping and distortion effects .......................................................................... 43
4.1 Design .......................................................................................................................................... 44
4.2 Finite Element Formulation ......................................................................................................... 46
4.3 Model Analysis ........................................................................................................................... 49
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4.3.1 Static Behaviour ................................................................................................................... 49
4.3.2 Dynamic Behaviour .............................................................................................................. 56
5. Deck's Dynamic Analysis .............................................................................................................. 61
5.1 Design Codes ............................................................................................................................... 62
5.2 Numerical Analyses .................................................................................................................... 64
5.3 Free Vibration Analysis ............................................................................................................... 66
5.4 Damped Forced Vibration Analysis ............................................................................................ 68
5.4.1 Application Example ............................................................................................................ 69
5.4.2 HSLM-A10 - Box girder Analysis ....................................................................................... 70
5.4.3 HSLM-A10 - Double-T section Analysis ............................................................................. 73
5.4.4 Concluding Remarks ............................................................................................................ 76
6. Concluding Remarks and Future Developments ........................................................................... 77
7. References ..................................................................................................................................... 80
8. Annex ............................................................................................................................................ 84
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List of Figures
Figure 2.1 - Relationship between damping ratio and frequency for Rayleigh damping,(Clough and
Penzien, 1995). ...................................................................................................................................... 14
Figure 3.1 - Longitudinal Model of simply supported beam under a vertical moving load .................. 20
Figure 3.2 - Cross Section Geometry considered in the practical examples ......................................... 23
Figure 3.3 - Dynamic influence line of vertical displacement at mid-span under a moving load ......... 25
Figure 3.4 - Dynamic influence line of vertical acceleration at mid-span under a moving load ........... 25
Figure 3.5 - Dynamic influence line of bending moment at mid-span under a moving load ................ 26
Figure 3.6 - Dynamic influence line of displacements at mid-span under a moving load with a)120ms-1
, b)260ms-1. ............................................................................................................................................ 27
Figure 3.7 - Dynamic influence Line of acceleration at mid-span under a moving load with a)120ms-1 ,
b)260ms-1. .............................................................................................................................................. 27
Figure 3.8 - Dynamic influence Line of bending moment at mid-span under a moving load with
a)120ms-1 , b)260ms-1. ........................................................................................................................... 28
Figure 3.9 - Dynamic influence line of displacement at mid-span under a moving load without
damping, (ξ= 0%) .................................................................................................................................. 30
Figure 3.10 - Dynamic influence line of acceleration at mid-span under a moving load without
damping,(ξ= 0%) ................................................................................................................................... 31
Figure 3.11 - Dynamic influence line of displacement at mid-span under a moving load with damping,
(ξ=2%) ................................................................................................................................................... 31
Figure 3.12 - Dynamic influence line of displacement at mid-span under a moving load with damping,
(ξ=2%) ................................................................................................................................................... 31
Figure 3.13 - Influence line of acceleration at mid-span by modal participation of each mode ........... 33
Figure 3.14 - Longitudinal Model of continuous beam under a vertical moving load .......................... 34
Figure 3.15 - Dynamic influence line of vertical displacement at mid-span of the first span under a
moving load ........................................................................................................................................... 36
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Figure 3.16 - Dynamic influence line of vertical acceleration at mid-span of the first span under a
moving load ........................................................................................................................................... 36
Figure 3.17 - Dynamic influence line of bending moment at mid-span of the first span under a moving
load ........................................................................................................................................................ 36
Figure 3.18 - Dynamic influence line of vertical displacement at mid-span of the first span under a
moving load with a)120ms-1 , b)260ms-1. .............................................................................................. 38
Figure 3.19 - Dynamic influence line of vertical acceleration at mid-span of the first span under a
moving load with a)120ms-1 , b)260ms-1. .............................................................................................. 38
Figure 3.20 - a) vertical displacement , b) vertical acceleration at mid-span of the first span over the
time due to 10 axle loads. ...................................................................................................................... 40
Figure 3.21 - a) vertical displacement , b) vertical acceleration at the mid-span of the first span over
the time due to 15 axle loads. ................................................................................................................ 40
Figure 4.1 - General Box crosssection geometry considered in the analysis. ....................................... 44
Figure 4.2 - General Double-T cross Section geometry considered in the analysis. ............................. 44
Figure 4.3 - 3D Frame model of the a) box cross section girder b) with dummy elements. ................. 48
Figure 4.4 - Load break down in symmetric and antisymmetric case for a box cross section .............. 50
Figure 4.5 - Load break down in symmetric and antisymmetric case for a double-T cross section ..... 50
Figure 4.6 - Reference point of analysis at middle span of the box-girder section and double-T-cross
section.................................................................................................................................................... 51
Figure 4.7 - Static displacement, direction ux, over the span of the a) box-girder section, b) double-T-
cross section due to the asymmetrical load. .......................................................................................... 52
Figure 4.8 - Static displacement, direction uy, over the span of the a) box-girder section, b) double-T-
cross section .......................................................................................................................................... 52
Figure 4.9 - Static displacement, direction uz, over the span of the a) box-girder section, b) double-T-
cross section. ......................................................................................................................................... 52
Figure 4.10 - Rotation, direction rx, over the span of the a) box-girder section, b) double-T-cross
section.................................................................................................................................................... 53
Figure 4.11 - Support conditions plan for a simply supported deck ...................................................... 54
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Figure 4.12 - Dynamic influence Line of normal stress at mid-span under a moving load, a) box-girder
section , b) double-T-cross section ........................................................................................................ 57
Figure 4.13 - Dynamic influence Line of displacement at mid-span under a moving load, a) box-girder
section , b) double-T-cross section ........................................................................................................ 57
Figure 4.14 - Dynamic influence Line of acceleration at mid-span under a moving load, a) box-girder
section , b) double-T-cross section ........................................................................................................ 58
Figure 5.1 - General model of the high-speed universal trains proposed by [EN 1991-2] .................... 62
Figure 5.2 - Longitudinal model of the continuous deck considered in the analyses. ........................... 65
Figure 5.3 - Cross section models with the eccentric load considered in the analyses. ........................ 65
Figure 5.4 - Dynamic influence lines of the vertical displacement at the midpoint of the central span.69
Figure 5.5 - Dynamic influence lines of the vertical acceleration at the midpoint of the central span. 70
Figure 5.6 - Envelope of maximum acceleration at midpoint of the central span according to the speed
............................................................................................................................................................... 71
Figure 5.7 - Envelope of maximum displacement at midpoint of the central span according to the
speed ...................................................................................................................................................... 71
Figure 5.8 - Envelope of maximum acceleration at midpoint of the central span according to the speed
............................................................................................................................................................... 72
Figure 5.9 - Envelope of maximum displacement at midpoint of the central span according to the
speed ...................................................................................................................................................... 72
Figure 5.10 - Envelope of maximum acceleration at midpoint of the central span according to the
speed ...................................................................................................................................................... 74
Figure 5.11 - Envelope of maximum displacement at midpoint of the central span according to the
speed ...................................................................................................................................................... 74
Figure 5.12 - Envelope of maximum displacement at midpoint of the central span according to the
speed ...................................................................................................................................................... 75
Figure 5.13 - Envelope of maximum displacement at midpoint of the central span according to the
speed ...................................................................................................................................................... 75
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List of Tables
Table 3.1 - Cross Section Properties considered in the practical examples .......................................... 23
Table 3.2 - Natural frequencies and vibration modes shapes ................................................................ 24
Table 3.3 - Analytical and Numerical natural frequencies of vibration of the box cross section.......... 30
Table 3.4 - Relative Error between the vertical acceleration of analytical analysis and numerical
analysis. ................................................................................................................................................. 32
Table 3.5 - Natural frequencies of vibration of the continuous beam. .................................................. 35
Table 3.6 - Maximum vertical acceleration at mid-span of the first span ............................................. 39
Table 4.1 - Geometrical and material properties of the cross sections considered in the analysis. ....... 45
Table 4.2 - Constrains applied to the shell element models .................................................................. 46
Table 4.3 - Longitudinal and Vertical Displacement at the first quarter of the beam of the simply
supported beam ..................................................................................................................................... 55
Table 4.4 - Natural Frequencies of vibration and respective vibration modes to each model of box
cross section. ......................................................................................................................................... 59
Table 4.5 - Natural Frequencies of vibration and respective vibration modes to each model of
thedouble-T cross section. ..................................................................................................................... 59
Table 5.1 - Dimensions and load magnitudes of the high-speed universal trains HSLM-A ................. 63
Table 5.2 - Definition of the set analysis cases and respective heights to perform the dynamic analysis.
............................................................................................................................................................... 66
Table 5.3 - Natural frequencies of vibration of the Box cross section. ................................................. 67
Table 5.4 - Natural frequencies of vibration of the double-T cross section. ........................................ 68
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List of Symbols
Latin Letters
𝐴- Cross section area
𝑐 - Damping coefficient
𝐶 - Global damping matrix
𝑑 - Distance between axles
𝐸- Young´s Modulus
𝑓- frequency
𝐹𝑒𝑥𝑐 - Excitation frequency
𝐼- Identity matrix
𝐼𝑦𝑦 - Moment of inertia around y axis
𝐼𝑧𝑧 - Moment of inertia around x axis
𝐽 - Torsion constant
𝑘 - Stiffness coefficient
𝐾 - Global stiffness matrix
𝐾𝑛 - Generalized stiffness associated to n-th mode of vibration
��- Effective stiffness matrix
m - Mass
𝑀- Global mass matrix
𝑀𝑛- Generalized mass associated to n-th mode of vibration
n - Integer, constant
𝑃𝑛- Generalized load associated to n-th mode of vibration
𝑃(𝑡)- Point load at time t
��- Effective loading vector
𝑡- Time
𝑢(𝑡)- Displacement
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��(𝑡)- Velocity
��(𝑡)- Acceleration
𝑣- Velocity of the moving loads
𝑉𝑐𝑟- Critical velocity
𝑤 - Circular frequency
𝑤𝑛 - n-th natural frequency of vibration
𝑤𝑏 - Circular frequency of damping
𝑥- Longitudinal space coordinate of a generic point
𝑦(𝑡) - Generalized coordinate vector
𝑌(𝑡) - Arbitrary normal coordinate
𝑌𝑛(𝑡)- Generalized displacement of n-th mode in time domain
Greek Letters
𝛾- Parameter of velocity control ( Newmark- β)
𝛽- Parameter of displacement control ( Newmark- β)
𝛿- Dirac delta function
𝜉𝑛 - Coefficients of the damping related to n-th mode of vibration
𝜃- Phase angle
𝜌- Mass per unit volume
𝛷- Mode shape of vibration
𝛷𝑛 - Mode shape of vibration associated to n-th mode
𝛷�� - Mode shape of vibration written for the normalized modal coordinates
𝜓��- Mode shape matrix written for the normalized modal coordinates
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1. Introduction
Over the years, the expansion of cities has led to the development of transportation infrastructures to
cope with the rise in demand. Railway services stand as one of the main transportation means in
suppressing those ever-increasing needs and today high-speed trains play a crucial role on the
circulation of people and goods.
Back in 1964, the Japanese built the very first high-speed railway line, called Tōkaidō Shinkansen.
Trains were designed to reach a running speed of 210 kmh-1. Short after the Shinkansen construction
began, the idea of a TGV arose in France. The first high-speed line in Europe, called South-East TGV
line, would start operating across France in 1981.
The development in Europe of high-speed trains in the recent past has introduced a new attention to
the study of vibrations induced in bridges by the rail traffic. The circulation of high-speed trains
introduces new load situations where dynamic effect become significant, especially in bridges, which
leads to the need of carrying out additional safety verifications, taking into account the checking of the
ultimate limit state and service in bridges, especially in resonance situations, an issue which has won
consequently an increased interest.
To attend to these effects, analysis procedures and verifications have recently been incorporated into
European standards that should be taken into account in bridges design, and involving aspects related
to structural safety and to passenger comfort.
Important torsional moments can be induced in the case of railway bridges crossed by eccentric
moving loads. Hence, the torsional response assumes a relevant role as well as the warping of the
cross section that cannot be neglected in analysis and design.
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1.1 The Aim of the Work
The present work aims to perform the dynamic analysis of multi-span high-speed railway bridges
under the eccentric loads using a numerical solution.
The modal analysis is considered to solve the equation of the dynamic equilibrium and obtain the
natural frequencies of the structures. The forced vibration is solved by using the modal superposition
method and the numeric integration, specifically the Newmark’s method.
To validate the numerical methods adopted in this work, an analytical solution to the dynamic analyses
under the moving loads, developed by Frýba [8], has been studied and implemented to obtain the
displacement of the simply supported beam and compared with the numerical solution of Newmark´s
method and with finite element method, developed in the commercial software SAP2000.
The finite element method is widely used in the dynamic analysis of structures. The use of numerical
finite element models for the development of this type of study raises a number of issues that matter to
be clarified in order to have more efficient and reliable procedures for the determination of the
dynamic effects on railway bridges for high-speed trains.
Therefore, an analytical solution to a simply supported beam is initially developed and then followed
on by a numerical solution to a continuous beam. Finally, an analysis of a multi-span bridge for two
different sections - Box section and Double T-section is performed. Some examples of those sections
under the eccentric moving loads are presented, in order to understand and evaluate the influence of
the distortion and the warping in the dynamic behaviour.
The dynamic analysis of bridges with a shell finite element model allows to obtain accurate results,
along with a detailed description. However, using such models at an early design stage might be
excessively time consuming, making the evaluation of multiple structural solutions cumbersome.
When numerous analyses with a set of moving loads and with great accuracy are necessary, the time
step becomes too small, and it can take too long to complete the dynamic analysis and obtain good
results. To overcome this, a shell finite element model is created to obtain the modes of vibration so
that the Newmark’s method can be applied in a spreadsheet.
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A vast analysis is carried out for different velocities and for different cross sections, to obtain an
envelope of the maximum acceleration and displacement of the bridge, in order to identify the
importance of resonance and of the different cross-sectional properties and deformations modes on the
dynamic behaviour of the bridge.
1.2 Literature review
Over a century, the study of dynamic behaviour of structures has been investigated, including the
study of bridges on railways subjected to moving loads. This theme has gained interest due to the fact
that on this type of structure, the dynamic displacement provided by a moving load is greater than the
maximum static displacement.
From the first constructions of railway bridges, many types of research have been developed to better
understand the dynamic analyse of the structures that are submitted to moving loads. Actually, this
knowledge is broadly improved.
The first studies about this subject were performed by Willis [26] in 1847 when following the colapse
of several bridges in England, the Queen determined the constitution of a commission whose
objective was the investigation of dynamic amplifications on bridges. On his studies, Willis
established the differential equation of a mass moving at a constant speed over a massless beam. The
simplification admitted conducted to the resolution of this problematic, however, it is considered a
first approach since this view does not take into account the inertial effects of the beam.
In 1922 Timoshenko [22] also performed a notable work on this theme. The author analysed the event
of a moving load or pulsating force passing over a simple beam, disregarding the inertia of the vehicle.
Furthermore, a solution for the event of a concentrated force moving with constant speed along a
prismatic bar, neglecting the contribution of damping forces, were established later by Timoshenko.
The scope of the study also included the critical speed and a formula was obtained.
More recently, with the publication of the text “Vibration of Solids and Structures under Moving
Loads”, Frýba [8] presents a huge contribution to the awareness of dynamic behaviour of structures
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under moving loads. The author presents an exhaustive analysis involving several cases, from one-
dimensional structures to three-dimensional solids. The train excitations were considered as moving
constant forces, harmonic forces or continuous loads. Moving forces randomly changing over the time
and moving multi-axle system were also taken into account, as well as a complete number of special
problems, including load motion at variable speed, random loads and forces moving at high-speed.
Frýba continued his research on the area of vibrations of railway bridges having the work resulted on
the publication of book “Dynamics of Railway Bridges” in 1996 [9]. On this compendium, the author
demonstrates a complete study about a significant number of parameters and simplifications, usually
assumed on the dynamic behaviour of theoretical railway bridges, and conclude about its influence on
these structures, either associated to bridge or to train models.
The influence of the speed of the train and of the track irregularities on the bridge response required a
special attention by Frýba. The problematic of fatigue on railway bridges was also considered.
Complementing the work, the experimental results were obtained, allowing the analysis between the
theoretical calculations and the measurements’ results.
In 2001, with the objective of understanding the resonance vibrations that can occur at critical speeds,
the Frýba worked in a theoretical model of a bridge. The author identified two different causes for the
resonance vibration on railway bridges: a) repeated action of train axle loads; and b) high speed of
moving loads by itself. This last cause for the vibration on railway bridges cannot be considered in
high-speed lines with the trains used nowadays. However, if repeated action of train axle loads were
applied, the resonance behaviour can occur, which result in unacceptable values of bridge deck
acceleration. This behaviour is considered the main reason for ballast destabilisation on small and
medium span bridges, as observed in some French bridges [10].
Due to the resonance vibration generated by the train axle load, special attention was done to this
phenomena, since a drastic amplification of the structural dynamic response can be created and
causing the ballasts destabilisation, in addition to excessive vibration in high-speed trains. By this
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reason, and for track maintenance reasons, it is recommended the mitigation of the problem from the
conception phase of the structure.
Supporting this rational, Yang et al. [23], performed a detailed study of the resonance phenomena,
modelling the actions according to the loads disposition. A train modelled composed by two
subsystems of wheel loads of constant intervals were considered, one referring to all the front wheel
assemblies and other regarding all the rear assemblies. It was identified the conditions for the
occurrence of the resonance and cancellation phenomena leading to the proposed optimal design
criteria that are effective for suppressing the resonant response.
Based on the studies performed it was possible to obtain conclusions, as the elongating of the period of
vibration of the beam by considering the inertial effects of moving vehicles.
The authors Y.B. Yang, J.D. Yau and Y. Wu also contributed to the development of this area. As
result from several studies performed previously, the authors compiled their achievements on the book
Vehicle-Bridge Interaction Dynamics: With Applications to High-Speed Railways [24]. A complete
analysis of the moving load problems (part 1) and vehicle-bridge interaction (part 2) is present through
the development of two and three-dimensional models, improving the numerical studies.
At the same time, Olsson (1991) [19], exhibited analytical and finite element solutions of a simply
supported beam, under a constant force moving at constant velocity. The analytical solution presented
by the author is based on a series solution, similar to the one given by Frýba [8].
In several articles have been discussed the beam vibration excitation by a moving mass. One of them
is from Michaltsos et al. [18] where was obtained a closed form solution for a single-span bridge under
the action of a moving mass of constant magnitude and velocity. A series solution for beam deflection
is derived applying the modal superposition method and considering as a first approximation the
solution of the corresponding vibration problem, disregarding the mass effect.
The possibility of separation of the moving mass from the beam during the motion was studied by Lee
[14] and [15], for Euler-Bernoulli and Timoshenko single span beams, respectively. On the other
hand, Zhu and Law [27] used Hamilton’s principle to determine the dynamic response of a continuous
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beam under moving loads. For this propose, the Newmark time integration method was been used and
high precision results have been achieved.
The reports on dynamic problems of multi-span bridges are limited. Yang et al., 1995 [25] proposed
impact formulas for vehicles traversing simple and continuous bridges. In turn, Lee, 1996 [16]
analysed the dynamic response of multi-span Euler-Bernoulli beams exposed to a moving mass. Also,
Ichikawa et al., 1999 [12] performed a similar study but disregarding the mass effect. However, using
the modal analysis, the effects of acceleration of the moving load were estimated on the impact factor
for a symmetric three-span beam.
1.3 Layout of the work
The present work is divided into six chapters sorted by the sequence of work adopted, excluding this
initial chapter where a general introduction to the current work is presented and contextualised in the
current panorama of high-speed rail. The motivations and developments needed are also illustrated.
In Chapter 2, an introduction to the dynamic behaviour of the structures, natural frequencies, mode
shapes and damping are presented to give support to the work developed in the subsequent chapters.
Also, the methodologies used to perform the dynamic analyses in which the loads vary over time, as
well as the Modal Superposition and Numeric Integration are introduced.
The study on moving loads is developed in Chapter 3, where the analytical solution developed by
Frýba is addressed. Some illustrative examples are developed to compare the analytical solution with
the numerical solution. It begins with a simply supported beam and follows through with a continuous
beam.
In Chapter 4 the modelling techniques adopted are approached. The cross section to be used in further
analyses is designed and the assumptions for the finite element models are explained, including some
discussion about the influence of the eccentric loads and the importance of the support conditions.
The main objective of the present work is developed in Chapter 5. A summary of standards and type
of trains used is firstly provided. Thereafter the analysis of the multi-span bridge due to the passage of
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high-speed trains acting eccentrically is carried out. Finally, the accelerations and displacements at the
mid-span are obtained for the different cases of analysis.
In chapter 6 the results obtained for the various analyses are discussed in order to understand the
differences between the different case scenarios. The main conclusions are presented and some ideas
are proposed for future developments.
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2. Dynamic Analysis
In this chapter, the basic concepts of Dynamics, used in the development of this work, are introduced
in order to provide the reader with the tools to understand the meaning of the dynamic equation in the
context of the analyses performed.
The analysis of dynamic behaviour begins with a formulation of equations of motion associated to
each dynamic degree of freedom of the structure, where for all instant is necessary to ensure the
equilibrium between external forces, that act in the direction of the displacement degree of freedom in
which the load is applied, and internal forces, that resist to motion, the inertial force, the damping
force and the elastic restitution force.
𝑚��(𝑡) + 𝑐��(𝑡) + 𝑘𝑢(𝑡) = 𝑃(𝑡) (2.1)
The equation (2.1) corresponds to the dynamic equilibrium equation. Where 𝑚 represents the mass of
one degree of freedom, 𝑐 the damping, 𝑘 the stiffness and 𝑃(𝑡) the external force. The displacement,
velocity and acceleration at time t are represented by 𝑢(𝑡), ��(𝑡) and ��(𝑡) respectively.
The solution of the differential equation of motion is obtained by the sum of the homogeneous solution
with the particular solution. The homogeneous solution is designated by the transient state since the
damping effect leads to a decrease in movement over the time. On the other hand, the particular
solution is designated by the steady state.
This is a problem with two points. From one hand, the free vibrations are analysed, based on the
natural frequencies of vibration and on the corresponding mode shapes, from another hand, the forced
vibrations are evaluated, and the structure’s response to a prescribed loading is determined, which can
also be obtained from the referred modal frequencies and shapes.
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2.1 Vibration Frequencies and Mode Shapes
Starting over simple structures with one degree of freedom, where mass 𝑚 is considered to be lumped.
The natural frequency is obtained by neglecting the damping effect and considering a null loading
applied, as follow:
𝑚��(𝑡) + 𝑘𝑢(𝑡) = 0 (2.2)
The vibration motion of the structure is harmonic with a given frequency 𝑤 , and may be expressed by
the solution of the equation of the motion:
𝑢(𝑡) = 𝐴𝑠𝑒𝑛(𝑤𝑡) + 𝐵𝑐𝑜𝑠(𝑤𝑡) (2.3)
Where 𝐴 and 𝐵 are the constants which depend on the initial conditions of the motion. Substituting the
solution on the equation (2.2) is possible to determine the natural frequency of the structure, which just
depends on the mass and stiffness of the structure:
𝑤 = √𝑘
𝑚 (2.4)
Actual structures are analysed with models with infinite degree of freedom since, in fact, they are a
continuous system. This reality is usually represented by a discrete model which permits to solve the
dynamic analysis by a discrete model with finite degree of freedom.
In that case, the equation of motion is composed by matrices of mass 𝑀, stiffness 𝐾 and damping 𝐶.
Hence a problem of eigenvalues and the eigenvectors is defined, where there is a system of dependents
differential equation.
Similar to the behaviour of one degree of freedom systems, it can be assumed that the free vibration
motion is simple harmonic, which may be expressed by the dynamic response as follows:
𝑢(𝑡) = 𝛷𝑠𝑒𝑛(𝑤𝑡 + 𝜃) (2.5)
11
The vibration mode shape of the system is represented by 𝛷, which is constant in time and 𝜃 is a phase
angle. Simplifying it is possible to write the solution considering 𝑦(𝑡) the term which depends on
time:
𝑢(𝑡) = 𝛷𝑦(𝑡) (2.6)
Replacing the solution (2.6) in the equation of motion, and since the sine term is arbitrary and may be
omitted, it can be written as follow:
−𝑤2. 𝑀. 𝛷 + 𝐾. 𝛷 = 0 (2.7)
The vibration mode shape 𝛷 and the correspondent natural frequencies 𝑤 can be computed from the
associated generalized eigen values problem, [3]:
[𝐾 − 𝑤2. 𝑀]𝛷 = 0 (2.8)
Where 𝑤2 and 𝛷 represent the eigenvalues and the eigenvectors, respectively. The eigenvalues
indicate the square of the freevibration frequencies, while the correspondent displacement vectors 𝛷
express the correspondent shapes of the vibrating system.
Meanwhile 𝛷 is non-trivial, that is, the displacement vectors 𝛷 are non-null, and therefore, the
following relation must be verified:
det(𝐾 − 𝑤2𝑀) = 0 (2.9)
There are so many modes of vibration 𝛷𝑛 as degrees of freedom and expanding the determinant will
give an equation of the n solutions 𝑤𝑛 of the n vibration modes. So, the n roots of this equation
represent the frequencies 𝑤𝑛 of the n modes of vibration which are possible in the system.
It is noted that the amplitude of the vibration mode is not defined. The resolution of the eigenvectors
and eigenvalues problem just allows to get the deformed configuration of each mode, but not its
amplitude. Hence, each modal vector can be normalised, and any criterion can be used for that
normalisation.
12
2.1.1 Modal Coordinates
A fundamental aspect of the modal analysis is the orthogonality of the vibration modes, a property that
is very useful in structural dynamic analysis. The orthogonality relations can be demonstrated by
application of Betti´s Law [4], which allow expressing the following conditions:
𝛷𝑚𝑇 𝑀𝛷𝑛 = 0 (2.10)
𝛷𝑚𝑇 𝐾𝛷𝑛 = 0 (2.11)
On the other hand, whether the same mode is multiplied by transpose mode relatively to mass or
stiffness matrix, it is obtained the generalised mass 𝑀𝑛 or stiffness 𝐾𝑛, respectively, which are
diagonal matrices.
𝑀𝑛 = 𝛷𝑛𝑇𝑀𝛷𝑛 (2.12)
𝐾𝑛 = 𝛷𝑛𝑇𝐾𝛷𝑛 (2.13)
Based on the orthogonal principles described above, it is possible to write the configuration of the
mode shape in other coordinates, modal coordinates. Considering as criterion, the normalisation of the
𝑛 mode by mass matrix 𝛷��, as follow:
𝛷�� =𝛷𝑛
√𝛷𝑛𝑇𝑀𝛷𝑛
(2.14)
Under these coordinates changes, it is possible to rewrite the motion equation in a simple way, since
with the orthogonal condition the following equations are obtained:
𝛷𝑛��𝑀𝛷�� = 1 (2.15)
𝛷𝑛��𝐾𝛷�� = 𝑤𝑛
2 (2.16)
Therefore, the mass matrix is transformed in the identity matrix and the stiffness matrix transformed in
a diagonal matrix in which each term of the diagonal is defined by the square of the correspondent
frequency of vibration.
13
Hence, by considering that the modal coordinates, the governing equations of motion can
be decoupled and a system of independent equations is obtained. The advantage of this procedure lies
in the fact that the characteristic matrices of the system, when referred to the modal coordinates,
become diagonal, allowing a separate analysis of each new degree of freedom.
2.2 Viscous Damping
The damping represents the way how structure can dissipate the energy, the reason why the dynamic
response decrease over the time.
There are several ways to formulate the damping matrix 𝐶, since the damping values may not be
constant for all the vibration modes, [4]. One way to quantify the damping properties, when damping
mechanisms are considered distributed throughout the structure, is by establishing a proportional
relation regarding the modal mass and stiffness properties.
Rayleigh demonstrated that if a matrix of damping is defined through the linear combination between
the mass and stiffness matrix, i.e.:
𝐶 = 𝑎0𝑀 + 𝑎1𝐾 (2.17)
then the vibration modes and frequencies are the same obtained for the undamped system.
So as it was shown earlier, the mass and stiffness matrix are orthogonal in relation to the vibration
modes which, through a linear combination, allow to determine the damping matrix to which the
vibration modes are also orthogonal.
The coefficients 𝑎0 and 𝑎1 could be obtained solving the system where two vibration modes are
considered, as follow:
[𝜉𝑚
𝜉𝑛] =
1
2[
1
𝑤𝑚𝑤𝑚
1
𝑤𝑛𝑤𝑛
] [𝑎0
𝑎1] (2.18)
Reformulating the system is possible to obtain:
14
[𝑎0
𝑎1] = 2 [
𝑤𝑛 −𝑤𝑚
−1
𝑤𝑛−
1
𝑤𝑚
] [𝜉𝑚
𝜉𝑛] (2.19)
Where 𝑤𝑛 and 𝑤𝑚 are the frequencies related to n-th and m-th vibration mode shape of the structure;
and 𝜉𝑛 and 𝜉𝑚 are the coefficients of the damping for each of the modes, respectively.
Looking at the systems 2.18 and 2.19, it is possible to conclude that for proportional damping to mass,
the damping factor is inversely proportional to the angular frequency and for the damping proportional
to stiffness, this damping factor is proportional to angular frequency.
The relationships between damping ratio and frequency can be seen in Figure 2.1.
Figure 2.1 - Relationship between damping ratio and frequency for Rayleigh damping [4].
2.3 Forced Vibration
The dynamic response of a linear system can be analysed through different methods. The way of
solving the equation implies the integration of the differential dynamic equation, which can be either
done by using the methods of modal superposition or by direct numerical integration. The mode
superposition method is based on reducing the system equation of motion in a number of independent
mode shapes that, despite being generally much lower than the number of dynamic mode shapes of the
structure, can represent the displacements with sufficient accuracy. The final response is then
calculated through the principle of superposition effects whereas the numerical integration consists of
the direct integration of the equation to obtain the history of the response in time through the use of
15
incremental procedures (step-by-step integration). Within this method, there are still some differences
in the resolution of the integration which differ in the approach and in the stability of the solutions
obtained. In this work, the numerical integration method chosen to perform the analysis was
Newmark's. In the following sections, a brief explanation of these two methods, modal superposition
and direct numerical integration, is presented as they will be applied in further analyses.
2.3.1 Modal Superposition
The Modal Superposition method´s consists in decoupling of differential equations by transforming
the initial coordinates in modal coordinates [4]. A system of linearly independent equations is created
using the orthogonality conditions of vibration modes, which makes possible studying each mode in
an independent way, greatly facilitating the resolution of the dynamic equations.
Using the orthogonal conditions and multiplying both sides of equation of equilibrium with ��T =
[𝛷1 𝛷2𝛷3
… . 𝛷��], which represents the matrix with the normalized vibration modes to the mass
matrix results in:
��T𝑀����(𝑡) + ��T 𝐶����(𝑡) + ��T 𝐾𝜓 𝑌(𝑡) = ��T 𝑃��(𝑡) (2.20)
Where it is possible to note the generalised mass of the beam as well as the generalised load and
consequently the equation can be simplified. Hence each independent equation can be expressed in a
simple form, as follows:
𝑀𝑛 ∙ ��𝑛(𝑡) + 𝐶𝑛𝑌��(𝑡) + K𝑛𝑌𝑛(𝑡) = 𝑃𝑛(𝑡) (2.21)
So, it is possible to transform the problem of n degrees of freedom into n problems with one degree of
freedom. Afterwards the resolution of these independent equations, each differential equation with
only one degree of freedom, can be solved by direct integration methods such as the integral of
Duhamel, the Fourier series, etc., and the solution can be obtained by the sum of each of the individual
contribution, through:
𝑢(𝑥, 𝑡) = ∑ 𝛷𝑛(𝑥)∞1 𝑌𝑛(𝑡) (2.22)
16
Where 𝛷𝑛and 𝑌𝑛 represents the n-th mode shape vibration and the modal amplitude respectively. The
modal amplitude is an important factor of each vibration mode that should be taken in account in order
to evaluate the contribution of each vibration on the dynamic response.
2.3.2 Numeric Integration
The integration of differential equations at discrete intervals of time is a rational way of obtaining an
approximate solution for problems where forces vary over space and time, since no analytical solution
can be found.
To perform the numeric integration of differential equations at intervals of discrete time, the equation
of dynamic equilibrium is rewriting this form:
𝑀∆�� + 𝐶∆�� + 𝐾∆𝑢 = ∆𝑃 (2.23)
The method based on the equation of motion and kinematics relation determines the response in terms
of displacements, velocities and accelerations of the system. There are different step-by-step analysis
methods which differ from each other in terms of the assumptions for the kinematic relations. The
approach using the step-by-step dynamic response analysis makes use of integration step to forward
from the initial to the final conditions for each time step.
Newmark´s Method
The Newmark´s Method is an implicit method for direct integration of the differential equations
system of dynamic equilibrium, which assume a constant acceleration along the time interval, so admit
a hypothesis of a linear variation of velocity into interval times ∆𝑡. The kinematic equations of
integration for speed and displacement in an instant 𝑡 + ∆𝑡 are the following:
𝑢𝑡+∆𝑡 = 𝑢𝑡 + (1 − 𝛾)∆𝑡��𝑡 + 𝛾∆𝑡��𝑡+∆𝑡 (2.24)
𝑢𝑡+∆𝑡 = 𝑢𝑡 + ∆𝑡𝑢𝑡 + (1
2− 𝛽) 𝛥𝑡2��𝑡 + 𝛽𝛥𝑡2��𝑡+∆𝑡 (2.25)
17
The parameters 𝛾 and β are the conditions of stability of the method. By Clough and Penzien [4],
𝛾 provides a linearly varying weighting between the influence of the initial and the final accelerations
on the change of velocity and the factor 𝛽 similarly provides for weighting the contributions of these
initial and final accelerations to the change of displacement. Adopting 𝛾 =1
2 and β =
1
4 this method
is unconditionally stable and termed as the constant average acceleration method at each time step.
The values of 𝑢𝑡+∆𝑡 and 𝑢𝑡+∆𝑡 can be obtained from these equations and after substituted in the
general equation of motion (2.23), since:
∆𝑢 = 𝑢𝑡+∆𝑡 − 𝑢𝑡 (2.26)
Rewriting the equations in the form of static problem, as follow:
��∆𝑢 = �� (2.27)
Where the effective modal stiffness and the effective modal loading are defined as:
�� =𝑀
𝛽𝛥𝑡2 +𝛾𝐶
𝛽𝛥𝑡+ 𝐾 (2.28)
�� = ∆𝑄 + (1
𝛽∆𝑡��𝑡 +
1
2𝛽��𝑡)𝑀 + (
𝛾
𝛽��𝑡 − ∆𝑡��𝑡(1 −
𝛾
2𝛽)) (2.29)
Defining the initial conditions of motion and the interval of time, it is possible to start the process and
find the solution for the equation by numeric integration.
Relatively to the discretization in time, it is, ingeneral, sufficient to use a time step that corresponds to
10 intervals for the smallest period of the considered vibration modes [7]. The standard proposes to
limit the analysis of vibration modes up to 30 Hz or 1.5 times the frequency of the fundamental mode:
∆𝑡 =1
10𝑓= 0.0033 𝑠𝑒𝑐 (2.30)
18
19
3. Dynamic Response of Beams under Moving Loads
The Dynamic Response of Beam due to a moving load has been an object of study over the last
decades. At the beginning, Timoshenko [22] developed an analytic solution for a moving load with
constant speed, developed for a simply supported beam assuming the theory of Euler-Bernoulli. Later,
Frýba [8] generalised and developed the problem for different types of structures and loads. In that
study, he introduced an important concept to analyse the problem- the critical speed. The critical speed
is associated to the fact of the passage of equally spaced loads can produce the maximum response
when the time intervals between the passages of two successive loads and periods of free beam
vibrations are equal or are multiples of each other.
The analytical methodology has the advantage of understanding the main principles that govern the
dynamic behaviour of the structure. However, for a beam with multiple spans, this theoretical
foundation becomes too complex, requiring some additional assumptions or using a numeric analysis
to determine the maximum response of the structure.
In the present work, the validation of the adopted numerical methodologies is obtained through the
comparison with the analytical solution, so it can be used in the subsequent chapters. An analytical
solution of a simply supported beam under a moving load is studied, being the influence of velocity or
damping analysed. These analyses proceed with the study of numerical solutions obtained with a finite
element model and the validation between numerical and analytical results is done. Finally, a
continuous beam is studied using numerical methodologies. For this element, the impact of having a
set of moving loads in lieu of a single one is studied, in order to gain a better understanding of this
effect since the ultimate goal of this work is analysing a continuous bridge under the loading of a high-
speed train.
20
3.1 Simple Supported Beam
To analyse the displacement produced by a moving load it is necessary to take into consideration
many factors. According to Frýba [8], the correct procedure requires the following assumptions:
The beam behaviour is described by a differential equation of Euler-Bernoulli;
The mass of moving load is negligible when compared with the beam´s mass;
The load moves with a constant speed;
The beam damping is proportional to the velocity of vibration;
The calculus will be carried through for a simply supported beam, so the beam has
zero deflection and zero bending moment at both ends.
Frýba deduces the following differential equation for the beams, where 𝛿 represent the Delta-Dirac
Function:
𝐸𝐼𝜕4𝑢(𝑥,𝑡)
𝜕𝑥4 + 𝑚𝜕2𝑢(𝑥,𝑡)
𝜕𝑡2 + 2𝑚𝑤𝑏𝜕𝑢(𝑥,𝑡)
𝜕𝑡= 𝛿(𝑥 − 𝑣𝑡)𝑃 (3.1)
Where 𝑥 and 𝑣 are the position and the velocity of the moving load, respectively, as can be observed
in the following figure:
Figure 3.1 - Longitudinal Model of simply supported beam under a vertical moving load.
By imposing the boundary conditions and the initial conditions, it is possible to obtain the solution of
the displacement by the methods of integral transformation, as it can be seen in the equation 3.2.
Therefore, the displacement, the velocity and the acceleration of an intended given point are known,
since the last two can be obtained from the differentiation of the displacement function.
21
𝑢(𝑥, 𝑡) = 𝑢0 ∗ ∑1
𝑗2[𝑗2(𝑗2 − 𝛼2)2 + 4𝛼2𝛽2
∞
𝑗=1
[𝑗2(𝑗2 − 𝛼2)sin (𝑗𝛺𝑡)
−𝑗𝛼[𝑗2(𝑗2 − 𝛼2) − 2𝛽2]
(𝑗4 − 𝛽2)2𝑒−𝑤𝑏𝑡 sin( 𝑤(𝑗)
´ 𝑡)
− 2𝑗𝛼𝛽(cos (𝑗𝛺𝑡) − 𝑒−𝑤𝑏𝑡𝑐𝑜𝑠(𝑤(𝑗)´ 𝑡)]sin (
𝑗𝜋𝑥
𝑙 )
(3.2)
where uo represents the static deflection at mid-span, wj the natural frequency of the j-th mode and wb
the frequency of damping, with 𝑤𝑏 = 𝑤𝑗√1 − 𝜉𝑗2, beeing 𝑤(𝑗)
´ obtained by 𝑤(𝑗)´ = √𝑤𝑏
2 − 𝑤𝑗2. The
dimensionless parameters, α and β, are characteristic of the effect of speed and damping,
respectively, 𝛼 =𝛺
𝑤1 and β =
𝑤𝑏
𝑤1 , where 𝛺 is the excitation frequency of the structure and is given by
𝛺 =𝜋𝑣
𝑙 .
An alternative solution can be considered through numerical methods based on finite elements which
permit to solve structures more complex. Numerically, the calculus for dynamic solicitations produced
by the passage of a train can be approached in two ways, through the direct integration over time and
by modal superposition, as explained previously insection 2.3.
Another important aspect to be considered in the analysis is the effect of resonance - the moving loads
can produce an important amplification of the dynamic response. This effect produces an increased
response of the structure when the excitation frequency and the structure natural frequency are the
same.
The excitation frequency of a set of equally spaced moving loads is given by:
𝑓𝑒𝑥𝑐 =𝑣
𝑑 (3.3)
where d is the distance between axles and v the velocity of the set of loads.
When the excitation frequency is close to one of the structures' frequency a resonance phenomenon is
induced.
22
In a real scenario, the train is simulated by a set of forces with similar spacing, moving along the track
line in the space with a constant velocity but with a variable position in the time. This means that the
dynamic action depends on the velocity of circulation of the train and on the excitation frequency that
can have a huge variation.
There is a possibility that the vibrations caused by successive axles running on a bridge can exceed the
effects of a single moving load travelling at the same time [7]. From that point of view, one can think
of a set of loads moving at a critical speed Vcr, whose excitation frequency is a multiple of the
vibration frequency, defined by:
𝑉𝑐𝑟 =𝑑𝑛𝑗
𝑖 (3.4)
Where nj is the natural frequency of vibration for mode j and i it is a multiple of the frequency of the
structure1 (i = 1, 2, 3...or i =1/, 1/3, ...).
3.1.1 Analytical Solution
In order to illustrate these moving loads effects previously introduced, the following example of a
simply supported beam is presented with the aim of evaluating its behaviour as well as having an
analytical solution, which validates the numerical methods used in further analyses.
The vibration modes, the vertical displacements and consequently, by derivation of the expression
3.2, the accelerations and the bending moments are analysed in this study. However, there are different
factors impacting on the structural response, as the speed the loads or the damping of the structure.
These two factors are studied in this section: the load speed, once because a certain range of speed can
lead to a situation where there is an amplification of the dynamic response and the structure undergoes
a resonance situation; the damping, to show the effect of the damping ratio on the dynamic response of
the structure.
23
The design considered for the analysis is based on the real properties of a simply supported bridge
deck, 40 meters long and with the cross section presented in Figure 3.2, which have been chosen to
follow on from previous works [17].
The cross-sectional properties are presented in Table 3.1, where for the purpose of this dissertation the
superimposed dead load was not taken into account.
Figure 3.2 - Deck Cross Section Geometry considered in the practical examples.
Table 3.1 - Cross Section Properties considered in the practical examples.
Property Box Section
𝑬 (𝒌𝑵/𝒎𝟐) 33 000 x 103
𝑮 (𝒌𝑵/𝒎𝟐) 13 750 x
103
𝝆 (𝒕𝒐𝒏/𝒎𝟑) 2.548
𝑨 (𝒎𝟐) 8.295
𝑰𝒚𝒚 (𝒎𝟒) 6.666
𝑰𝒛𝒛 (𝒎𝟒) 104.145
The first six frequencies and respectively vibration modes that were obtained, using the equation (2.4),
are presented in Table 3.2:
24
-1
0
1
0 40
-1
0
1
0 40
-1
0
1
0 40
-1
0
1
0 40
0
1
0 40
-1
0
1
0 40
Table 3.2 - Natural frequencies and vibration modes shapes.
Mode
Number
Frequency
Hz
Mode Shape
Mode
Number
Frequency
Hz
Mode Shape
Analytical Analytical
1
3.332
4
53.311
2
13.328
5
83.301
3
29.998
6
119.95
To obtain the analytical solution for a moving load, the Frýba solution (3.2) was used. The solution
depends on the speed and damping, which are represented by the dimensionless parameters α and β,
which for some special cases, as critical damping or no damping, can be neglected and the equation
simplified. Also, by working out this equation the solution for the vertical acceleration and bending
moment can be obtained.
Influence of the velocity
To have a better understanding of the velocity effect of a moving load on a beam behaviour, several
analyses considering different velocities were performed. Five velocities were considered (80 ms-1,
120 ms-1, 200 ms-1, 260 ms-1, 300 ms-1). Despite being higher than the range of train velocities, these
values were chosen to take into account the effect of resonance, since the critical speed of the simply
25
supported beam in study can be achieved for that range of velocities. The damping was neglected and
the amplitude of the moving load P is 1000 kN.
The results are presented in Figures 3.3, 3.4 and 3.5, for the displacement, acceleration and bending
moment at mid-span of the beam, respectively.
Figure 3.3 - Dynamic influence line of vertical displacement at mid-span under a moving load.
Figure 3.4 - Dynamic influence line of vertical acceleration at mid-span under a moving load.
-10
-5
0
5
10
0 10 20 30 40 50 60 70 80
Dis
pla
cem
en
t [
mm
]
x [m]
80 m/s 120 m/s 200 m/s 260 m/s 300m/s
forced vibration free vibration
uo
-8
-4
0
4
8
0 10 20 30 40 50 60 70 80
Acc
ele
rati
on
[m
s-2 ]
x [m]
80 m/s 120m/s 200 m/s 260 m/s 300 m/s
forced vibration free vibration
amax=7.68 ms-2
26
Figure 3.5 - Dynamic influence line of bending moment at mid-span under a moving load.
As it can be observed, the maximum values of deflections are approximately the same for different
load speeds, around 10 mm, although the time at which they occur is different. An exception can be
notice to the velocity of 80 ms-1 since considering a lower velocity the deflection is smaller and its
value is close to the static deflection u0, represented by the dashed line. In that case, the increase of the
effect due to a dynamic load is given by the Dynamic Amplification Factor, 𝐷𝐴𝐹 =𝑢𝑚𝑎𝑥
𝑢0, which is
smaller when comparing to the other higher velocities.
In relation to acceleration, higher values are generally associated with higher velocities, in spite of the
peak acceleration occurring at a speed of 260 ms-1. This can be explained by the fact that the period of
excitation, 𝑇 =2L
v , is approximately the same as the period of the structure, which means that 260 ms-1
cause a resonance phenomenon, leading to higher responses. Relatively to the bending moment, it can
be concluded that different velocities do not change the maximum value of stress resultants, only the
instant at which the structure is loaded. In addition, it can be also noticed that the stress resultants are
proportional to the vertical displacements.
-15,000
-10,000
-5,000
0
5,000
10,000
15,000
0 10 20 30 40 50 60 70 80
Be
nd
ing
Mo
me
nt
[kN
m]
x [m]
80 m/s 120m/s 200m/s 260m/s 300m/s
forced vibration free vibration
27
Influence of Damping
In terms of the damping analysis, and to understand what is the impact on the structure, two examples
with different velocities were used, one with the lower velocity (120 ms-1) and another with the higher
velocity (260 ms-1). Three different types of damping, 0 %, 1% and 2% are considered.
The results can be observed on the graphics presented in Figures 3.6, 3.7 and 3.8 and they show the
influence line of the vertical displacement, acceleration and bending moment at the mid-span of the
beam for the different coefficients of damping.
a) 120ms-1 b) 260ms-1
Figure 3.6 - Dynamic influence line of displacements at mid-span under a moving load with a)120ms-1, b)260ms-1.
-10
-5
0
5
0 10 20 30 40 50 60
Dis
pla
cem
en
t [m
m]
x [m]ξ = 0% ξ = 1% ξ = 2%
-10
-5
0
5
10
0 20 40 60 80
Dis
pla
cem
en
t [m
m]
x [m]
ξ = 0% ξ = 1% ξ = 2%
a) 120ms-1 b) 260ms-1
Figure 3.7 - Dynamic influence line of acceleration at mid-span under a moving load with a)120ms-1, b)260ms-1.
-8
-4
0
4
8
0 20 40 60 80
Acc
ele
rati
on
[m
s-2]
x [m]
ξ = 0% ξ = 1% ξ = 2%
-8
-4
0
4
8
0 20 40 60 80
Acc
ele
rati
on
[m
s-2]
x [m]ξ = 0% ξ = 1% ξ = 2%
28
Analysing the results obtained, it can be concluded the damping effect does not influence the case of
resonance (260 ms-1) neither the case with lower velocity (120 ms-1). In regard to the displacements, in
both cases, it is possible to observe that the damping does not virtually change the vertical deflection.
The same conclusion can be drawn when the acceleration results are analysed, since it can be observed
the damping effect does not change the behaviour of the structure when is loaded, just during the free
vibration being the damping influence noticed. Finally, in terms of internal forces, the maximum
moment does not differ, despite the way the structure is loaded being different.
In bridge structures, the damping mechanism is highly complex. Examination of test results on
individual structures indicates that damping varies for different frequencies and amplitudes of
oscillation [7]. The results obtained on this sample showed the same level of response for different
damping ratios, although disparate results could have been obtained for a difference range of speeds.
3.1.2 Numerical Solution
The numerical analysis methods are mainly based on finite element method. There are currently
several commercial programs that allow performing the analysis by this method. To use the
a) 120ms-1 b) 260ms-1
Figure 3.8 - Dynamic influence Line of bending moment at mid-span under a moving load with a)120ms-1, b)260ms-1.
-10,000
-5,000
0
5,000
10,000
15,000
0 20 40 60 80
Be
nd
ing
Mo
me
nt
[kN
m]
x [m}
ξ = 0% ξ = 1% ξ = 2%
-10,000
-5,000
0
5,000
10,000
15,000
0 20 40 60 80
Be
nd
ing
Mo
me
nt
[kN
m]
x [m]
ξ = 0% ξ = 1% ξ = 2%
29
programme for numerical calculus is necessary to have a total knowledge about itself, in such a way
that the results can be reliable.
To perform the numerical analysis it was used the commercial program SAP2000, in order to verify
their reliability and therefore enable a comparative study between the numerical results from the
program and the results obtained in a beam of Euler-Bernoulli, using the analytical solution developed
by Frýba. Additionally, the numeric integration considering the Newmark method was also considered
to do the comparative study since it can be another quick way to get the numerical results. This
method was supported by the finite element model since the frequencies and the vibration modes were
calculated through this program and then exported to the spreadsheet where the numerical method of
Newmark was performed. Thus, three analysis methods were compared:
Analytical
Numerical - Time-History Analysis
Numerical - Newmark Method
Considering these methods, a validation of the numerical results was done which allow to develop
models of a continuous beam and to have reliability one its results. There are also several important
factors which need to be taken into account,as the mesh size adopted in the finite element program and
the importance ofconsidering each vibration mode, to obtain the final dynamic response of the
structure, which is discussed in this work.
Validation of the numerical results
To understand the importance of the numerical methods and how results can be obtained faster than
through an analytical method, a finite element model composed of frame elements as well as the
Newmark Method were developed. The same section properties used in the analytical solution were
considered, Figure 3.3, which allow to compare the response of both solutions. This structure was
modelled with 1 meter length frame elements and it was adopted a time increment of 0.002 sec to
perform the numeric integration.
30
The simply supported beam with 40 meters span was subjected to a single load of 1000 kN moving at
120 ms-1. Two cases of study were considered, the first one without damping and the second one
considering 2% damping ratio.
Firstly the natural frequencies obtained through the analytical solution and the numerical solution were
compared, which can be observed in Table 3.3, afterwards a forced vibration analysis was performed.
The values of the structure’s displacement and acceleration using Frýba solution, the Newmark
method and doing the analyses in SAP2000, can be observed between Figures 3.9 and 3.12.
Table 3.3 - Analytical and Numerical natural frequencies of vibration of the box cross section.
Natural Frequencies of Vibration - Hz
Frequency Number 1 2 3 4 5 6
Model of
Analysis
Analytical 3.332 13.328 29.988 53.312 83.300 119.952
Numerical 3.316 13.217 29.576 52.975 82.878 118.657
Figure 3.9 - Dynamic influence line of displacement at mid-span under a moving load without damping, (ξ= 0%).
-10
-5
0
0 10 20 30 40
Dis
pla
cem
ent
[mm
]
x [m]
Analytical Fryba Numerical Newmark Numerical Finite Element
31
Figure 3.10 - Dynamic influence line of acceleration at mid-span under a moving load without damping,(ξ= 0%).
Figure 3.11 - Dynamic influence line of displacement at mid-span under a moving load with damping, (ξ=2%).
It is important to note that it was enough to considered six modes shapes to reach accurate results,
which confirm that analyse by modal superposition it is a good way to achieve the correct response
-2
3
0 10 20 30 40
Acc
eler
atio
n [
ms-2
]
x [m]
Analytical Fryba Numerical Newmark Numerical Finite Element
-10
-5
0
0 10 20 30 40
Dis
pla
cem
ent
[m
m]
x [m]
Analytical Fryba Numerical Newmark Numerical Finite Element
Figure 3. 12- Dynamic influence line of acceleration at mid-span under a moving load with damping, (ξ=2%).
-2
-1
0
1
2
3
0 10 20 30 40
Acc
eler
atio
n [
ms-2
]
x[m]
Analytical Fryba Numerical Newmark Newmerical Finite Element
32
with less numerical effort, since it allows with just six modes describe the behaviour of a structure
with many degrees of freedom. This effect can be observed in Figure 3.13 for the case of the
displacements at mid span. Simultaneously, for the numeric integration and considering the increment
of the time, it gives an accurate response of the structure.
However, there is an important factor that needs to be considered when the finite elements method is
used, since the element refinement may have interference in the results. Considering this, three
different meshes were considered with 40, 80 and 120 frame elements, which were used to perform the
analyses for the undamped case of study.
The results presented in Table 3.4 shows that a rough mesh does not impact the accuracy of the results
since the numerical results using the discretization of 120 and 80 elements are extremely close to the
results when 40 elements are used. This allows saving computational time, getting results accurate
enough comparatively to the analytical results.
Table 3.4 - Relative Error between the vertical acceleration of analytical analysis and numerical analysis.
Maximum Vertical Acceleration at Mid-Span
Model of Analysis Analytical
Frýba
Finite Element
Nr. Frame Elements
120 80 40
Acceleration mid span [ms-2] 2.423 2.407 2.398 2.396
Error (%)
=𝐀𝐧𝐚𝐥𝐲𝐭𝐢𝐜𝐚𝐥 − 𝐍𝐮𝐦𝐞𝐫𝐢𝐜𝐚𝐥
𝐀𝐧𝐚𝐥𝐲𝐭𝐢𝐜𝐚𝐥
- 1.6% 2.5% 2.7%
As mentioned previously, to understand the number of vibration modes that should be taken into
account in the analysis and their relevance on the structure final response, the same example presented
above was analysed, again not considering the damping effect.
33
The graphic presented, Figure 3.13, shows the influence line of acceleration at mid-span obtained
when the structure is analysed with only one up to six vibration modes. It can be observed that, whilst
the first mode has the highest modal participation, the second, fourth and sixth modes do not have any
contribution at all, being the third and fifth the secondary modes contributing to the final response.
Those factors can be seen in the side table, where the modal participation is showing the influence of
each vibration mode.
Figure 3.13 - Influence line of acceleration at mid-span by modal participation of each mode.
3.2 Continuous Beam
Nowadays, the bridgesdeck are not usually simply supported structures but rather a continuous
structure, which arouses interest in the study of this type of structures.
Chan and Ashebo [2] developed a method to get the exact mode shape function of the vibrating beam
with intermediate supports. However, as the number of spans of the bridge increases, the identification
accuracy decreases and at the same time, more execution time is needed to finish one case study. So,
due to the characteristics and complexity of the differential equation, the numerical solution was
carried out.
In this section is studied the dynamic behaviour of a continuous beam through the finite element
solution and Newmark Method. First, a single moving load was applied to the continuous beam in
-2
-1
0
1
2
0 10 20 30 40
Acc
ele
rati
on
[m
s-2 ]
x [m]
1 mode 2 modes 3 modes
4 modes 5 modes 6 modes
Mode
Number
Modal
Participation
Maximum
acceleration
x=20m
1 83% 1.913
2 0% 1.913
3 9.155% 2.244
4 1.32E-18% 2.244
5 3.24% 2.396
6 1.99E-15% 2.396
34
order to obtain the results of the forced vibration in terms of displacements, acceleration and bending
moment. Furthermore, a study of the same beam supporting a set of a concentrated moving loads was
performed, considering the superposition of the dynamic responses of each single load to obtain the
final response The influence of having two or many moving loads considering different cases of
analysis was studied, each analysis considered a different delta Δ, space between loads, to understand
the importance of considering that effect since the final goal is to model a real train composed by a set
of loads.
3.2.1 Continuous beam under concentrated load
A continuous beam with two spans subject to a single moving load was analysed, as represented in
Figure 3.14. The length of each span and the section properties were considered the same as in the
examples above at section 3.1. It was possible to study the behaviour of the structure by numerical
analyses since the reliability of the numerical studies discussed previously are known, which are
necessary to obtain accurate results.
To perform the analysis a frame model by finite element was developed. The single moving load
applied to the continuous beam had the amplitude of 1000 kN, the damping was ignored and five
different cases of velocities were studied.
Figure 3.14 - Longitudinal Model of continuous beam under a vertical moving load.
In Table 3.5, the first six frequencies and their respective vibration modes obtained using the finite
elements can be observed:
35
-1
0
1
0 20 40 60 80
-1
0
1
0 20 40 60 80
-1
0
0 20 40 60 80
-1
0
1
0 20 40 60 80
-1
0
1
0 20 40 60 80
-1
0
1
0 20 40 60 80
Table 3.5 - Natural frequencies of vibration of the continuous beam.
The results due to the forced vibration cases are presented from Figure 3.15 to 3.17. The displacement,
acceleration and bending moment of the structure were measured at the mid-span of the first span in
order to have a better understanding and to allow the comparison with the previously analysis with one
span.
Mode
Number
Frequency
Hz
Mode Shape
Mode
Number
Frequency
Hz
Mode Shape
Numerical Numerical
1
3.269
4
14.919
2
4.948
5
25.828
3
12.405
6
28.484
36
Figure 3.15 - Dynamic influence line of vertical displacement at mid-span of the first span under a moving load.
Figure 3.16 - Dynamic influence line of vertical acceleration at mid-span of the first span under a moving load.
Figure 3.17 - Dynamic influence line of bending moment at mid-span of the first span under a moving load.
-12
-8
-4
0
4
8
12
0 20 40 60 80 100 120 140 160
Dis
pla
cem
en
t [m
m]
x [m]
80 m/s 120 m/s 200 m/s 260 m/s 300 m/s
-8
-4
0
4
8
0 20 40 60 80 100 120 140 160
Acc
ele
rati
on
[m
s-2]
x [m]
80 m/s 120 m/s 200 m/s 260 m/s 300 m/s
-20,000
-10,000
0
10,000
20,000
0 20 40 60 80 100 120 140 160
Be
nd
ing
Mo
me
nt
[kN
m]
x [m]
80 m/s 120 m/s 200 m/s 260 m/s 300 m/s
37
By analysing the results for different velocities, it was possible to verify that when the load is moving
over the beam, the displacements are very close but after that, the structure in free vibration has a
bigger deflection for higher velocities. In terms of acceleration, by comparing the results obtained for
a single span, it can be observed that there is no peak for a load running at 260 ms-1, the conditioning
velocity for a single span. One reason for that fact can be explained by the mode shape of vibration of
the different structures where for a continuous beam the third mode of vibration does not have as
much importance at mid-span of the first span as the simply supported beam. Consequently, the
structure has less movement at this point and so the acceleration does not increase since there is no
contribution for that. In relation to the forces that can be observed when the load is moving over the
beam, it is noted that the bending moment is greater when the velocity is lower, although these
differences are not significant.
3.2.2 Continuous beam applied for a set of concentrated loads
To realise how a real train acts, it is important, rather than a concentrated load, to analyse the effect
that many loads can have in the structure. It is possible to obtain the results by superposition of the
response for each load action. The aim of the following analyses was to demonstrate how the structure
responses change when they were subjected to a set of concentrated loads instead of a single point
load.
The structure was subjected to two concentrated loads, being possible to obtain the results through the
superposition of the response for each load action as a single load on the beam. The analysis was done
for two speeds, 120 ms-1 and 260 ms-1, and each load had 500 kN. For each speed, different cases of
analysis were considered, according to the spacing, Δ, between the loads. Finally, these analyses were
compared with the previous example with a single load with 1000 kN, where Δ was zero.
The dynamic influence line of the displacements and acceleration at first half span is presented in the
Figure 3.18 and 3.19:
38
Table 3.6 presents the maximum accelerations obtained for each case that shown in Figure 3.19.
a) 120ms-1 b) 260ms-1
Figure 3.19 - Dynamic influence line of vertical acceleration at mid-span of the first span under two moving loads with a)120ms-1 ,
b)260ms-1.
-8
-4
0
4
8
0 40 80 120
Acc
ele
rati
on
[m
s-2]
x [m]-8
-4
0
4
8
0 50 100 150
Acc
ele
rati
on
[m
s-2]
x [m]
a) 120ms-1 b) 260ms-1
Figure 3.18 - Dynamic influence line of vertical displacement at mid-span of the first span under two moving loads with a)120ms-1
, b)260ms-1.
-12
-6
0
6
12
0 50 100 150
Dis
pla
cem
en
t [m
m]
x [m]-12
-6
0
6
12
0 50 100 150
Dis
pla
cem
en
t [m
m]
x [m]
39
Table 3.6 - Maximum vertical acceleration at mid-span of the first span.
Maximum Acceleration [ms-2]
Load Type
1000kN 500 kN 500kN
Distance between
loads Δ =0 m Δ =10m Δ =13.33m Δ =20m Δ=40m
120 [ms-1] 1.71 0.96 0.83 0.74 1.09
260 [ms-1] 7.68 6.9 6.2 4.5 3.21
By analysing the values in Table 3.6 it can be concluded that the effect of a concentrated load was
much higher than the effect of two point loads, with half load each, and the larger was the spacing
between loads the less was the displacement and the acceleration caused by the load. However, in the
particular case when considering the case analysed for 120 ms-1 with two loads spaced at 40 meters,
the acceleration of the continuous beam increased. The conclusion that can be taken is the fact of the
critical speed being present, as explained previously in equation 3.4, the loading was moving with a
speed close to the natural frequency of vibration multiplied by the spacing between axles. In that
specific case, with 120 ms-1, the critical spacing was 36.36 meters, 𝑑 =𝑉𝑐𝑟
𝑛𝑗, leading to an amplification
of the dynamic response of the structure which depends on the number of loads, i.e., the duration of
resonance, and the intensity of the loading.
The importance of considering different loadings is understood by the fact that the superposition of
effects of two moving loads separated in time, can lead to the amplification of the dynamic response.
Hence, analyses considering 10 and 15 axle loads are carried out.
This study aims to comprehend the importance and the influence in simulating the loading train
instead of a single concentrated load. To perform these analyses, it was considered a load of 500 kN
per axle, moving, with 120 ms-1 and a spaced of 36.36 meters between each load since it is the critical
space achieve through the previous analysis.
The vertical displacements and acceleration at first half span over the time can be observed in the
Figures 3.20 and 3.21:
Δ
40
By analysing the figures 3.20 and 3.21, one can conclude that the displacements and accelerations
increase significantly when a set of loads with the critical space is considered instead of just one or
two loads with the same spacing. The maximum vertical displacement was around 10 mm in both
cases and the maximum vertical acceleration was 6 ms-2 and 8 ms-2 in the first and second cases,
respectively. So, the acceleration was more than two times higher than the obtained in the previously
case, with just one or two moving loads, for the same velocity. It can also be concluded that the
a) b)
Figure 3.21 - a) vertical displacement , b) vertical acceleration at the mid-span of the first span over the time due to 15 axle loads
moving at 120 ms-1.
a) b)
Figure 3.20 - a) vertical displacement , b) vertical acceleration at mid-span of the first span over the time due to 10 axle loads
moving at 120 ms-1.
-12
-8
-4
0
4
8
12
0 1 2 3 4 5 6
Dis
pla
cem
en
t [m
m]
Time [s]
-8
-4
0
4
8
0 2 4 6
Acc
ele
rati
on
[m
s-2]
Time [s]
-12
-8
-4
0
4
8
12
0 1 2 3 4 5 6
Dis
pla
cem
en
t [
mm
]
Time [s]
-6
-3
0
3
6
0 2 4 6
Acc
ele
reta
ion
[m
s-2]
Time [s]
41
acceleration values reach the same level when the analysis was done with higher speeds, 260 ms-2
which means, for this type of analysis, that it was not the velocity that controls the high values achieve
for the structure but the passage of successive loadings.
42
43
4. Deck Modelling - warping and distortion effects
The focus of this work is to study the influence of non-conventional deformation modes (eg.
distortion, warping) in the analysis of bridge deck under the effect of moving loads. The distortion
results in displacements in the plane of the cross section, whereas the warping effect introduces
longitudinal out-of-plane displacement. These effects can significantly change the dynamic behaviour
of the structure leading to important changes in the concept design phase of a bridge. These analyses
are performed through numerical models, whose reliability has been discussed in Chapter 3.
To evaluate the transverse distribution of the effects induced by high-speed trains along the bridge
cross section, it is useful to compare the behaviour of two different cross sections, one open section, a
double-T beam, and a box girder with a much higher torsional stiffness. The two types of cross
sections are analysed and also different heights of each one are put in comparison.
Finite element models were developed in which the bridge’s deck was modelled by shell and frame
finite elements. The shell model allows the distortion and warping to occur, whereas the frame
element is not able to capture these effects by virtue of its kinematics. The comparison between results
obtained with the two models becomes a useful method of assessing the influence of the distortion and
warping on the behaviour of the structure.
Additionally, two other different factors were analysed: the eccentricity of the load, that may influence
the local effects of the cross section, and the position of the supports, which can affect the response of
the structure . The study of these factors becomes easier in the context of a static analysis, as the
interaction with the dynamic effects can make it seem overly complex. Hence, the study of non-
conventional deformation modes begins through the static analysis, and then it is finalised with the
dynamic analysis.
44
4.1 Design
The cross sections considered in this work allow the comparison between two possible structural
solutions for the bridge design, and have been chosen from the solutions adopted in previous works
[21] on related subjects, where the superimposed dead load was not taken into account. The geometry
of the cross section is defined by orthogonal webs and flanges in both cases and has been chosen in
order to have approximately the same elastic properties (i.e. area and second moment of area in the
flexure plan) to allow the comparison of both solutions and the understanding of the effect of the
torsion, as there is a significant difference between the behaviour of the two cross sections. In order to
understand the influence of the cross section stiffness on the dynamic behaviour of the bridge, three
different heights for each cross section were analysed.
The general geometry definition of the cross sections used in further analyses is shown in Figure 4.1
and 4.2. The mechanical properties of the cross sections are presented in Table 4.1 as function of the
height h.
Figure 4.1 - General Box cross section geometry considered in the analysis.
Figure 4.2 - General Double-T cross section geometry considered in the analysis.
45
Table 4.1 - Geometrical and material properties of the cross sections considered in the analysis.
Properties Box Section Double-T Section
Case h1 Case h2 Case h3 Case h1 Case h2 Case h3
h (m) 1.68 2.18 2.68 2.53 3.03 3.53
𝑨 (𝒎𝟐) 7.695 8.295 8.685 8.566 9.506 10.166
𝑰𝒚𝒚 (𝒎𝟒) 4.190 6.666 11.120 6.133 9.161 14.5623
𝑰𝒛𝒛(𝒎𝟒) 96.275 104.145 109.6192 111.992 122.371 129.687
𝒛𝑪𝑮 (𝒎) 0.496 0.670 0.853 0.588 0.772 0.968
𝒛𝑪𝑪 (𝒎) 0.653 0.817 0.975 0.499 0.663 0.840
𝑱 (𝒎𝟒) 10.857 15.416 24.860 0.977 1.224 1.319
𝑬 (𝒌𝑵/𝒎𝟐) 33 000 x 103 33 000 x 103
𝑮 (𝒌𝑵/𝒎𝟐) 13 750 x 103 13 750 x 103
𝝆 (𝒕𝒐𝒏/𝒎𝟑) 2.548 2.548
As it can be seen in Table 4.1, as the section becomes deeper the second moment of area and the mass
of the section increase. From equation 2.4, it follows that changes in the stiffness and in the mass of
the structure will impact on its natural frequency and, consequently lead to a different dynamic
behaviour. The choice of a box girder and a double-T section allows to measure how much impact the
torsional stiffness has on the dynamic response of the bridge. In fact, whilst the bending stiffness and
the cross-sectional area of these two sections are in the same order of magnitude, the ratio between
their torsional constants, J, is in the order of ten.
46
4.2 Finite Element Formulation
As explained previously, one of the objectives of this work is to analyse the influence of the cross
section distortion in the overall behaviour of the bridge. To that end, it is necessary to develop a model
that permits isolating that phenomenon in such a way it can occur freely or be fully restrained, so that
its influence on the structural behaviour can be assessed. A finite element program is used to analyse
models corresponding to different types of hypothesis of the cross section behaviour, simulated
through constraints between displacements which is a particularity that can be defined when using a
shell element.
A constraint consists of a set of two or more constrained joints used to enforce their displacements to
relate. This can be useful to simulate different behaviours, e.g. rigid along the axis or, more
importantly in the context of this work, rigid in the plane of the cross section. In order to achieve a
structural response with no distortion, a body constraint was needed to be defined along the bridge at
each cross section. This type of constraint causes all of the constrained joints to move together as a
sectional in-plan rigid body. Effectively, all constrained joints are connected by rigid links and cannot
displace relatively to each other.
Initially, two different shell models were developed. The first one does not have any degree of
freedom constrained over the cross section, which allows the cross section to deform freely and
allowing to have free warping and distortion. The second is a model with some constraints in order to
restrain the cross section distortion. The restraint corresponding to each of the referred models can be
seen in Table 4.2.
Table 4.2 - Constrains applied to the shell element models.
ux uy uz өx өy өz
Free Warping and Distortion - - - - - -
Restraining Distortion - x x x - -
47
The shell element always considers all six degrees of freedom at each joint of the element. Having no
constraints in the displacements and rotations of a cross section allows local effects, e.g. warping and
distortion, to occur. In order to prevent the distortion effect, it is necessary to restrain the relative
displacements on the YZ plane of the cross section and the rotation about X axis. However, by
applying constraints to a model using shell elements one can only expect these phenomena to be
partially restrained, once the warping can still occur. Thus, a third model that intrinsically does not
allow these local effects to happen by virtue of its kinematics became necessary. A frame element
model was then developed as it utilises Bernoulli’s beam formulation where plane section remains
plane, that is to say, no warping or distortion are considered.
In this way three different finite element models were developed to the box girder and double-T cross
sections, each one with three different heights:
Shell model allowing distortion and warping
Shell model restraining distortion
Frame model
For the purpose of studying a bridge under the hypotheses of no warping and no distortion, a frame
element model was developed as opposed to shell-element one. By doing so, numerical issues,
associated with the use of rigid elements constraining these kinematic effects, were avoided. The use
of a simpler finite element formulation, not capturing warping or distortion in its kinematics, provides
a model computationally more efficient and also more reliable.
Shell Model
Shell elements are used to capture the three-dimensional behaviour of structures because, as opposed
to frame elements, they allow multiple joints in one single cross section to move out of their plane,
allowing the section to lose its planarity (i.e. warping), and to rotate on the cross section plan causing
its inner angles to vary (i.e. distortion). All finite elements adopted in the models are quadrilaterals and
48
have a dimension of approximately 1 meter on both sides. This type of model allows to obtain the
dynamic response at any point of the bridge deck and therefore to understand the local effects, which
cannot be achieved with simplified models.
The mass contribution from the shell element is lumped at the element joints and a thin-shell type is
considered, which means that it follows the Kirchhoff formulation, where the transverse shear
deformation is neglected.
In relation to the support condition, it was adopted a discrete support located at the centroid of the
cross section, which restrains the transversal displacements on all supports and the longitudinal
displacements only on the left abutment support. No rotation at the supports has been restrained.
Finally, as explained previously, body constraints were applied at different sections along the deck to
simulate the behaviour of the cross section girder.
Frame Model
The frame model is a simple way to represent the bridge deck. It is composed by a set of frame
elements, linking joints within a straight alignment, as shown in Figure 4.3 a). The mechanical cross
sections corresponding to the cross section defined by shell elements were assigned to the frame
elements.
As for the discretization, the frames were assumed with the approximate 1 meter length. The
parametric study on the meshing carried out in Chapter 3 led to the conclusion that frame elements of
1 meter could be used for accurately simulating the bridge under the action of moving loads.
a) b)
Figure 4.3 - 3D Frame model of the a) box cross section girder b) with dummy elements.
49
The software assumes automatically the joints at the centroid of the frame cross section, once each
element is modelled as a straight line connecting two joints. This means that the loads can only be
applied at that point and that to apply an eccentric moving load it is necessary to define dummy frame
elements (elements with no mass to ensure that loading transfers without the object affecting the
response of the structure) with the distance of the respective eccentricity and then connect these back
to the structure by means of transverse dummy elements, Figure 4.3 b).
The supports were defined at the centroid and restrain the transverse displacements on all supports and
the longitudinal displacements only on the left abutment support.
4.3 Model Analysis
Upon defining the deck section geometry, some analyses were performed to understand how different
constraints reducing warping and distortion of the cross section impact on the bridge global behaviour.
Thus, three combinations of constraining were tested:
Distortion and warping not constrained;
Distortion constrained and warping not constrained;
Distortion and warping constrained.
These analyses consider the fact that the deck is loaded with an eccentric moving load in relation to its
centroid. Finally, a static analysis has been performed since the analyses without dynamic effects can
be easier to understand the load path and the displacements of the structure. Then dynamic analyses
with a single moving load were performed to allow the understanding of how sections behave when
submitted to a real train load.
4.3.1 Static Behaviour
The study of the local effects becomes easier in the context of a static analysis, as the interaction with
the dynamic effects can make it seem overly complex.
50
Since the models are created using different assumptions and particularities, it is necessary to find a
way to understand the results obtained. In order to do so, a static analysis was performed. In a first
stage, each model is eccentrically loaded allowing to understand how the structure responds and to
observe each local effect. In the second stage, different support conditions are considered to see how
important it is to correctly model the boundary conditions and to analyse its consequences on the
structural response.
Eccentricity Influence
The positioning of each track in two-way lines is eccentric in relation to the centroid, which introduces
a torsion effect and an important transverse deformation of the cross section. To understand this
problem it is necessary to decompose the eccentric load into symmetric and antisymmetric
components as shown in Figures 4.4 and 4.5.
The symmetric part causes longitudinal bending and the antisymmetric yields two systems, one with
torsion deformation, in which the section twists like a rigid body, and other with distortion. Thus, by
combining the different static systems due to the eccentric loading, three types of displacement are
obtained: vertical displacement due to longitudinal bending, rotation and distortion.
Figure 4.4 - Load break down in symmetric and antisymmetric case for a box cross section.
Figure 4.5 - Load break down in symmetric and antisymmetric case for a double-T cross section.
51
The main objective is to study the influence of the different cross section when the same load cases are
applied, being expectable to have higher displacement at double T-cross section. On the other hand, it
is also an objective to understand the distortion effect of the cross section and the importance of the
different section constraints when the eccentric load is considered.
The analysis is done considering the case h2 properties of both cross sections, defined in Table 4.1, in
a deck simply supported at the centre of gravity and with 40 meters length, as previously used for
other cases of study. The eccentric load is considered above the web, with 200 kN/m and it is broken
down into cases with 100 kN/m as can be seen in the Figures 4.4 and 4.5. All the support restraints are
considered at the centroid of the cross section.
The structure response is obtained by the superposition of the responses for each load case, symmetric
and antisymmetric. As explained previously the symmetrical load case causes bending moment, which
results in only vertical displacements with no local effects of distortion and warping to be analysed.
Thus, Figures 4.7 to 4.10 show the response due to the asymmetrical load case allowing the analysis of
the cross section distortion and warping effects. The results are taken at the middle span of the cross
section girder, Figure 4.6 shows the specific point over the cross section girder, where the presented
displacements were evaluated.
Figure 4.6 - Reference point of analysis at middle span of the box-girder section and double-T-cross section.
Figures 4.7 to 4.10 show the longitudinal, transverse and vertical displacements and the rotation about
the longitudinal axis obtained through the analysis of the asymmetrical load case for the simply
supported beam.
52
a) Box cross section b) Double-T cross section
Figure 4.8 - Static displacement, direction uy, over the span of the a) box-girder section, b) double-T-cross section due to the
asymmetrical load.
a) Box cross section b) Double-T cross section
Figure 4.7 - Static displacement, direction ux, over the span of the a) box-girder section, b) double-T-cross section due to the
asymmetrical load.
0
0.2
0.4
0.6
0 10 20 30 40
Dis
pla
cem
en
t u
y [m
m]
x [m] -8
-6
-4
-2
0
0 10 20 30 40D
isp
lace
me
nt
uy
[mm
]
x [m]
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 10 20 30 40
Dis
pla
cem
en
t u
x [m
m]
x[m] -0.8
-0.4
0
0.4
0.8
0 10 20 30 40
Dis
pla
cem
en
t u
x[m
m]
x [m]
a) Box cross section b) Double-T cross section
Figure 4.9 - Static displacement, direction uz, over the span of the a) box-girder section, b) double-T-cross section due to the
asymmetrical load.
-6
-3
0
0 10 20 30 40
Dis
pla
cem
en
t u
z [m
m]
x [m] -30
-20
-10
0
0 10 20 30 40
Dis
plc
ace
me
nt
uz[
mm
]
x [m]
53
Observing the Figure 4.7, it becomes clear that the frame element does not present longitudinal
displacements 𝑢𝑥 over the span since the model does not take into account warping. However, this can
occur in the shell models, where the double-T model shows higher displacements compared to the box
section, approximately 2.5 times. On the other hand, the distortion effect, in terms of relative
displacement, is more important for the box section than for the double-T section, which can be seen
comparing the result obtained when a shell model has the distortion restrained with the model
allowing warping and distortion. This effect can also be observed in Figure 4.8, which can be
explained by the higher torsional stiffness of the box section comparing to the double-T section and
also by the position of the shear centre. Figure 4.9, shows the vertical displacements obtained for both
sections which are much higher on the double-T section, 5 times higher than the box section. In the
frame model a significant difference can be seen on the double-T section results compared to the shell
models. Therefore, it can be concluded that the frame elements can represent, with good
approximation, a box section since the torsional stiffness is the main responsible for restraining the
movement while the double-T section resistance is given by the warping stiffness which is not present
on a frame element and, consequently, its displacements are larger than in a shell element. Finally,
Figure 4.10 illustrates the rotation of the cross sections along the longitudinal axis, reflecting what
have been presented. The double-T section shows higher rotation compared to the box section. In the
a) Box cross section b) Double-T cross section
Figure 4.10 - Rotation, direction rx, over the span of the a) box-girder section, b) double-T-cross section due to the
asymmetrical load.
-7.E-04
-2.E-04
0 10 20 30 40
Ro
tati
on
rx
[ra
d]
x [m]-9.E-03
-6.E-03
-3.E-03
2.E-17
0 10 20 30 40
rota
tio
n r
x [r
ad]
x [m]
54
case of the box section the rotations are approximately the same regardless of whether the distortion
and warping are both restrained or just the distortion is restrained.
Supports Influence
In order to make the model more realistic, a model with two supports located at bottom flange is
developed instead of one supported at the centroid. These supports are infinitely rigid since in this
work the abutment and analysis of the deformability of the soil are not considered, being the analysis
of the bridge deck the only object of study.
In order to depict more accurately the effect of the bearings, these have been placed at their actual
position, i.e. at the bottom flange underneath the webs, as vertically restrained joints. Once the frame
elements are defined at the centroid of their cross section, rigid frame elements have been used to
connect the centroid of the deck down to the supports. Since the bridge deck is the only object of this
study, an infinite stiffness was assigned to the supports. The abutments and the analysis of the
deformability of the soil are out of the scope.
The axial stiffness of the deck is significantly higher than the bending stiffness of the piers, which
means that the displacement of the deck at the top of each pier can be assumed to be free in the
longitudinal direction. Two supports per section are considered, and these restrain the displacement in
the transverse direction and permit the longitudinal displacement, except the left-hand supports, where
the longitudinal displacement is also restricted. The restraints are represented in Figure 4.11:
Figure 4.11 - Support conditions in-plan view for a simply supported deck
55
To analyse the influence of the supports, the results between the models with one support at the
centroid and two supports at the bottom of the cross section have been compared. In the same way, a
40m long simply supported beam was analysed for an eccentric load of 200kN/m for both support
arrangement. The study includes the box girder section and the double T-cross section respectively,
with different section restraints.
It is possible to verify in Table 4.3 the results for different cases, where the displacement was taken at
the first quarter of the beam in the point shown in Figure 4.6.
Table 4.3 - Longitudinal and Vertical Displacement at the first quarter of the span of the simply supported beam.
Displacement
(mm)
Shell model allowing
distortion/warping
Shell model
restraining distortion
Frame
Model
Box - Cross section
1 Support
ux– Longitudinal 1.51 1.42 0
uz– Vertical 27.65 24.34 22.58
2 Supports ux - Longitudinal 5.18 4.88 3.39
uz - Vertical 26.91 24.27 22.58
Double-T-Cross Section
1 Support
ux - Longitudinal 1.45 1.40 0
uz– Vertical 25.78 23.81 38.93
2 Supports ux - Longitudinal 5.52 5.26 3.98
uz– Vertical 21.65 20.46 38.55
As it can be observed, when an analysis is performed using a single support at the centroid of the
section, the vertical displacement is slightly higher than in the case where two supports restraining the
bottom flange are considered. The biggest difference is noticed in the case of the shell model
considering the double T section whereas in the frame model these differences are nearly null.
Regarding the longitudinal displacements, it can be observed that these increase in both sections when
two supports are considered. This effect is a consequence of the location of the supports, since the
distance from the top of the section, where the displacements are measured, is increased. Consequently
56
considering the support at the centroid leads to a lower displacement than considering the supports at
the bottom of the section. Another aspect worth noting is the importance of the different types of
constraints along the deck, i.e. warping and distortion can either be considered or neglected which can
change the longitudinal displacements. Once the section has no restraints it can warp, slightly
increasing the longitudinal displacement. Besides that this displacement is also different according to
the support conditions, as can be observed through the values presented in Table 4.3. Hence, the
accuracy of the results strongly depends on the way the supports are considered.
4.3.2 Dynamic Behaviour
The present section deals with the study of the dynamic effects of the different finite element models
developed in the previous section, where several factors changing the behaviour of the structure have
been noticed.
In the current analysis, the dynamic response of the structure is analysed with a longitudinal 40m long
bridge and with the section properties of case h2, as defined in Table 4.1, considering the different
cross section constraints defined previously, in order to compare the accelerations and displacements
between the different models. The support conditions are located at the centroid of the cross sections.
The structures are applied with a moving load of 1000 kN moving at 120 ms-1 with an eccentricity of
3.75m and 3.325m, to the box-girder and double-T cross section respectively, that is to say over the
web of each section, as shown in Figures 4.4 and 4.5. Also, the shell model allowing distortion and
warping is loaded at the centre of the cross section , to understand the effect of an eccentric loading
compared to a centred one.
These results were obtained at the mid-span, over the web, for an undamped beam considering
vibration modes with frequencies up to 30 Hz, assumed to be the frequency threshold above which the
vibration modes are considered to be negligible for the structure response [6], and can be seen in
Figure 4.12 to Figure 4.14.
57
a) Box cross section b) Double-T cross section
Figure 4.12 - Dynamic influence Line of normal stress at mid-span under a moving load, a) box-girder section , b) double-T
cross section.
-3
-2
-1
0
1
2
0 20 40 60 80
Ten
sio
n σ
x [M
pa]
-3
-2
-1
0
1
2
0 20 40 60 80
Ten
sio
n σ
x [M
pa]
a) Box cross section b) Double-T cross section
Figure 4.13 - Dynamic influence Line of displacement at mid-span under a moving load, a) box-girder section , b) double-T
cross section.
-15
-10
-5
0
5
10
0 20 40 60 80
Dis
pla
cem
en
t u
z [m
m]
-15
-10
-5
0
5
10
0 20 40 60 80
Dis
pla
cem
en
t u
z [m
m]
58
From the results presented in Figures 4.12 to 4.14, it becomes clear that the evaluation of the response
of the structure might be under conservative when a frame model is considered to represent a structure
with lower torsional stiffness. It is evident that the results of the double-T cross section have a
difference comparing to the shell models. Open cross sections tend to be particularly susceptible to
this effect since the warping stiffness represents the primary source of torsion resistance. Dynamic
displacements are, therefore, significantly affected by this consideration.
It also can be observed that the maximum vertical displacement, Figure 4.13, is obtained in the shell
model allowing warping and distortion, as this model has a higher flexibility than the others. On the
other hand, the maximum value of acceleration is obtained with the shell model restraining the
distortion. In regard to normal stress values, it can be observed that these values are proportional to the
vertical displacement obtained for each model.
Finally, in relation to the effect of the eccentric loading compared to the centred one, it can be
observed that the acceleration results obtained for theses two cases are very close in both cross
sections. However the stresses and the displacements are significantly different, being the effects of
the eccentricity in the double-T section higher than in the box cross section.
The first five vibration modes and respective frequencies are shown in Table 4.4.
a) Box cross section b) T cross section
Figure 4.14 - Dynamic influence Line of acceleration at mid-span under a moving load, a) box-girder section , b) double-T cross
section.
-5
-3
-1
2
4
0 20 40 60 80
Acc
ele
rati
on
uz
[ms-2
]
-4
-2
0
2
4
0 20 40 60 80
Acc
ele
rati
on
uz
[ms-2
]
59
Table 4.4 - Natural Frequencies of vibration and respective vibration modes to each model of box cross section.
Box Cross Section - Natural Frequencies of Vibration [Hz]
Frequency Number 1 2 3 4 5
Shell Model allowing
warping/distortion
3.02 7.99 10.02 10.08 13.28
Vertical
Bending Torsional
Vertical
Bending Lateral Bending Torsional
Shell Model neglecting
distortion
3.09 10.75 11.05 11.19 21.72
Vertical
Bending
Torsional/
Lateral
Bending
Vertical
Bending Torsional
Vertical
Bending
Frame Model
3.10 11.11 11.78 24.52 34.78
Vertical
Bending Lateral
Bending Vertical
Bending Vertical
Bending Lateral
Bending
Table 4.5 - Natural Frequencies of vibration and respective vibration modes to each model of the double-T cross section.
Double-T Cross Section - Natural Frequencies of Vibration [Hz]
Frequency Number 1 2 3 4 5
Shell Model allowing
warping/distortion
3.37 3.92 4.24 7.75 10.06
Vertical
Bending Torsional Axial
Torsional/
Lateral Bending Lateral Bending
Shell Model
neglecting distortion
3.44 4.02 4.26 11.05 12.41
Vertical
Bending Torsional Axial
Torsional/
Lateral Bending Torsional
Frame Model
3.46 10.3 13.25 28.03 29.69
Vertical
Bending Lateral
Bending Vertical
Bending Vertical Bending Lateral Bending
In conclusion, apart from the first vertical vibration mode, the other vibration modes of each model
demonstrate clear differences due to the constraints imposed at each cross section and consequently
different responses of the structure. Table 4.4 and Table 4.5, show the mode shapes classification
associated with the major modal participation of each section girder model. The different responses,
obtained for each model, clearly demonstrate how important it is to take local effects, as distortion and
warping, into account in the design process.
60
61
5. Deck's Dynamic Analysis
The main objective of this chapter is to analyse the characteristics of the dynamic answer of multi-
span bridges, related to an eccentric high-speed universal train into different bridge´s sections to
realise the distortion influence.
The rail traffic with high velocity could give rise to an excessive vibration of the structure. The
dynamic behaviour of bridges with two tracks and the eccentric passage of rail traffic implies the
existence of vibrations coupled of bending-torsion which can lead to an important deformation of the
transversal section.
Firstly, to perform the dynamic analysis of a multi-span bridge, a brief review of the codes to define
tracks, trains and velocities is made. Thereafter, tri-dimensional deck models are developed
considering the same cross sections used in the previous chapters, a box girder and a double-T section.
Through the deck modelling, the natural frequencies and mode shapes are obtained and then the forced
vibration due to an eccentric high-speed train is analysed with Newmark´s method.
Following on from the study developed in Chapter 4, the distortion effects were analysed with shell
models considering different types of constraints between displacements and compared with the
results of a frame model. These models were used on multi-span bridge models to perform a dynamic
analysis to understand the influence of non-conventional deformation modes. In this chapter, the
dynamic analyses are also performed with different heights for each cross section providing analyses
with different stiffness,as previously done in Chapter 4.
Finally, the dynamic study of a multi-span bridge subjected to a high-speed universal train is done for
a range of velocities which allows obtaining an envelope of maximum acceleration and displacement
to each model analysed and understanding the importance of the local effect on the bridge behaviour.
62
5.1 Design Codes
The railway regulation has evolved over the years since the trains have arisen. Into the European
space, the CEN –Comité Européen de Normalisation has developed codes and other guidances, in
order to standardise the design of structures being a result of this work the Eurocodes. The present
work follows these references, where is approached the loads to be considered in the design of railway
bridges as well as the procedures to be adopted to account the dynamic effects due to passage trains, in
addition to the necessary verifications to confirm the safety.
Characteristics of the trains
To performe a dynamic analysis, the EN1991-2 [6] norm prescribed the use of two high-speed
universal trains, HSLM-A and HSLM-B. This last one is just used for spans with less than 7 meters
while the first one is used for larger spans. At the present work, the analyses are done taking into
account the high-speed universal trains HSLM-A.
The HSLM-A is subdivided in 10 vehicle types to describe the loads model, denominated by HSLM-
A1 to A10, which were created to represent an envelope of existing trains in the European space, in
relation to a dynamic response. These trains differ from one to another in the numbers of carriages, in
the characteristic size, the spacing between the axle bogies and axle loads respectively, as can be
observed in Table 5.1 and Figure 5.1.
Figure 5.1 - General model of the high-speed universal trains proposed by EN 1991-2 [6].
63
Table 5.1 - Dimensions and load magnitudes of the high-speed universal trains HSLM-A [6].
HSLM-A
Universal
Train
Number of
Intermediate
Carriage
Carriage
Length
Spacing Between
The Axis Bogie
Force per
Axle
N D [m] d [m] P [kN]
A1 18 18 2 170
A2 17 19 3.5 200
A3 16 20 2.0 180
A4 15 21 3.0 190
A5 14 22 2.0 10
A6 13 23 2.0 180
A7 13 24 2.0 190
A8 12 26 2.5 190
A9 11 26 2.0 210
A10 11 27 2.0 210
The real trains, as well as the load models used, can be modelled by concentrated loads moving upon
the structure. Considering this hypothesis, the interaction effects between vehicles/structure are
neglected not being necessary to characterise and model the vehicle suspensions in terms of geometry,
mass, stiffness and damping.
Vertical Acceleration Criteria
In the EN1990-A2 [5] and EN1991-2 [6], it is defined a set of rules and limits related to the dynamic
response of railway bridges for high-speed lines. One of the limitations is the imposition of a
maximum vertical acceleration to avoid the train instability and the reduction of the wheel-rail contact
forces, ensuring the stability of lane and the safety of the railway movement. In ballasted bridges the
maximum vertical acceleration allowed is 3.5 ms-2 and in not ballasted bridges is 5.0 ms-2.
64
.
The velocity at which the vehicle is moving can be identified as a parameter of extreme importance in
the dynamic study. At speeds lower than 200 km.h-1, the vibratory phenomenon occurs generally with
small amplitude, but at higher velocities, the dynamic effects increase considerably and this fact can
be extremely amplified by using long trains. So, it is necessary a complete study about the dynamic
behaviour of the bridge that includes, in addition to the usual structural analyses, the research of
critical velocities where a high dynamic amplification can occur.
According to EN 1991-2 [6], it should be made a dynamic analysis with the lower velocity of
140km.h-1 (≈ 40 ms-1) and the maximum limit is defined by 1.2 times the maximum design speed
which means the maximum velocity of the lane at the bridge location, as defined in the project and
depends on the vehicles or structure characteristics.
The factor 1.2 provides a safety margin in relation to uncertainties in determining the natural
frequencies of the structure and consequently the resonant speeds.
In this work, it is considered 350 km.h-1 as the maximum design velocity and so the dynamic analysis
is performed up to a velocity of 420 km.h-1.
5.2 Numerical Analyses
In the present chapter, dynamic analyses are performed for an eccentric high-speed train moving over
two different cross sections, a box girder and a double-T section. The geometry and properties to each
case of cross section were described in section 4.1. In order to study the effect of the cross section
distortion on the dynamic analysis, two shell models were developed, one allowing warping and
distortion, other just restraining distortion, and a frame model that intrinsically does not allow warping
and distortion. All the constraint conditions mentioned were developed in detail insection 4.2.
The finite element program SAP2000 is the software used to develop all the finite elements models
since it has in-built tools to performing dynamic analysis with the characteristics mentioned
previously. However, there are some limitations on its operation, particularly when consuming
65
unacceptable calculation times and when performing analyses with integration steps for the required
accuracy to the high-speed area. To overcome this problem, a numerical analysis was done using
Newmark´s method implemented with macros developed in Visual Basic for Application (VBA) to
optimise the process.
To perform the dynamic analysis the vibration mode shapes were extracted from finite element
models, with a mesh size of approximately 1 metre, and were adapted to the time step of each velocity
by creating a macro which allows to interpolate the values, as shown in Annex A1. Once the values of
the vibration modes in the correct points are obtained, they can be used in the Newmark´s method. To
automate the process of considering the entrance of each single load and perform the Newmark´s
method, another macro was developed (refer to Annex A2) which allows to optimise the process and
achieve the results.
The deck dynamic analyses are performed for a three-span bridge. The dimension of each span are
presented in Figure 5.2 and the supports conditions considered are located under the web of the cross
section. Figure 5.3 have represented the eccentric load in the z-direction, which is considered with 2.5
meters distance of the centre of gravity to represents the eccentric path of the train.
Figure 5.2 - Longitudinal model of the continuous deck considered in the analyses.
Figure 5.3 - Cross section models with the eccentric load considered in the analyses.
66
First, it is analysed the undamped free vibration of each section considering frequencies up to 30 Hz,
as referred in EN 1990-A2 [5], followed by analysis of that damped forced vibration, where the
maximum displacement and speed of the section can be observed. All these analyses are done to
considering the cross section properties presented in Table 5.2, the remaining properties of these
sections were presented in detail in Chapter 4, section 4.3.
Table 5.2 - Definition of the set analysis cases and respective heights to perform the dynamic analysis.
Cross Section Analysis Cases
Box cross section - hi (m) Double-T cross section -hi (m)
Finite Element Model h1 h2 h3 h1 h2 h3
Shell Model allowing warping/distortion
h1= 1.68 h2= 2.18 h3=2.68 h1= 2.53 h2= 3.03 h3=3.53 Shell Model restraining distortion
Frame Model
5.3 Free Vibration Analysis
The free vibration analysis results of the two considered decks performed using the numerical analyses
can be observed in Table 5.3. The natural frequencies calculated for different cases of constraint can
be compared and demonstrated the clear influence of distortion and warping on the dynamic response.
During the analysis it is considered the frequencies for each section up to 30Hz, as mentioned
previously, this means that when a section is analysed considering warping and distortion there are
more degrees of freedom and local modes, therefore more modes to be taken into account up to 30Hz.
This was an important factor to be considered in this work as it can express the importance of local
modes on the dynamic behaviour.
In Table 5.3 it is possible to compare the first 12 natural frequencies obtained by the different models
of the box section cross section. It can be noticed that, in every model, the first frequencies of the
shell models are similar, corresponding to the first four flexure modes, while the fifth mode is a
torsional mode, with higher frequency in the model which restrain distortion, since it has constraints,
67
that make the model stiffer. The model allowing warping and distortion is more flexible leading to
local modes of vibration, as can be observed for the closest values of vibration. By virtue of not having
local modes of vibration, frame models need fewer modes of vibration to perform analysis with
frequencies of up to 30 Hz.
Table 5.3 - Natural frequencies of vibration of the Box cross section.
Natural Frequency Hz - Box Section
Mode
Number
Shell Model allowing
warping/distortion
Shell Model restraining
distortion
Frame Model
h1 h2 h3 h1 h2 h3 h1 h2 h3
1 3.06 3.85 4.60 3.12 3.95 4.72 3.19 4.02 4.78
2 4.58 5.45 5.98 4.70 5.59 6.09 5.01 5.92 6.38
3 5.30 6.48 7.56 5.46 6.74 7.93 5.89 7.17 8.38
4 7.45 8.08 8.86 7.71 8.44 9.38 8.64 9.21 10.21
5 7.57 8.38 8.89 9.38 11.34 12.99 11.38 14.00 13.84
6 9.20 10.08 10.74 10.31 12.86 13.49 14.16 14.10 16.55
7 9.47 10.80 11.69 12.47 13.58 15.2 17.05 20.61 20.65
8 9.68 11.23 12.40 12.72 15.11 17.37 17.96 20.99 22.45
9 12.51 13.39 13.26 13.74 15.38 17.63 21.33 21.66 24.69
10 12.82 13.59 13.97 15.19 18.73 19.86 22.66 25.25 25.03
11 12.91 13.72 14.03 15.85 19.33 21.09 25.75 25.86 27.67
12 13.53 14.34 14.48 19.25 20.16 22.49 25.83 27.73 31.70
The same conclusion can be taken by comparing the values obtained for the double-T cross section
which is presented in Table 5.4. Nevertheless, the fact of being an open section, with less torsional
stiffness, leads to more torsional modes of vibration and a greater difference between models. This
difference can be easily understood by the shapes of vibration since the frame model does not have
distributed mass along the cross section and consequently, cannot have torsional modes of vibration.
68
Table 5.4 - Natural frequencies of vibration of the double-T cross section.
Natural Frequency Hz - Double-T Section
Mode
Number
Shell Model allowing
warping/distortion
Shell Model restraining
distortion
Frame Model
h1 h2 h3 h1 h2 h3 h1 h2 h3
1 3.56 4.24 4.80 3.65 4.35 4.95 3.75 4.53 5.29
2 4.31 4.78 5.12 4.45 4.91 5.26 6.06 7.29 8.47
3 4.97 5.511 5.87 5.10 5.64 6.09 7.13 8.53 8.99
4 6.31 7.19 7.29 6.54 7.49 8.10 8.99 8.99 9.85
5 6.47 7.40 7.61 6.71 7.74 8.89 12.81 12.44 12.10
6 7.53 7.94 8.30 7.91 8.77 9.41 13.42 15.95 17.26
7 8.29 8.16 8.35 8.67 9.66 10.75 18.45 17.83 17.83
8 8.51 8.97 8.71 12.82 12.93 13.01 19.20 18.48 18.30
9 10.00 9.51 9.04 12.97 14.48 15.72 20.23 23.94 26.98
10 10.10 9.80 10.31 13.04 15.42 17.64 21.49 25.24 27.35
11 11.52 11.44 10.95 18.47 18.51 18.56 26.98 26.98 28.63
12 11.73 12.27 12.11 18.81 19.39 19.61 27.38 30.82 29.62
5.4 Damped Forced Vibration Analysis
This section aims to analyse the structural dynamic response of a three-span bridge subjected to high-
speed trains and investigate the local effects by considering different modelling assumptions. The
analyses are based on the codes defined previously in section 5.1, which follows the norms defined in
EN1991-2 and the use of models of high-speed universal trains HSLM-A.
The dynamic behaviour is investigated for a set of different velocity values of the moving loads,
ranging from 140 kmh-1 to 420 kmh-1, in accordance with EN 1991-2 (2003), considering an interval
of 20 kmh-1 between the velocities analysed. The high-speed load model considered in further analyses
is the HSLM-A10, with the magnitude of 210 kN axle load and the respective load disposition as
described in Table 5.1.
69
To understand the numerical integration, the modal superposition is used with a time step of 0.002 sec
and a modal damping of 1 % was considered.
In the present section dynamic analyses will be developed for the different cases of cross sections with
the aim of determining the maximum values of displacement and acceleration which allow creating an
envelope of maximum values according to the speed.
5.4.1 Application Example
To illustrate what has been done in further analyses to obtain the maximum values, Figures 5.4 and 5.5
provide an example of the dynamic influence lines of the vertical displacements and acceleration,
respectively, of the continuous deck represented in Figure 5.2.
The given example considers the high-speed train model HSLM-A10 to move at a constant speed of
200 kmh-1 over the box girder, considering the section with 2.18 m of height (h2), represented in Table
5.2, and the supports located under the webs of the cross section. The mode shapes of vibration were
taken from the finite element program and then used in a spreadsheet prepared to perform dynamic
analyses with the Newmark´s method as explained in section 5.2. The results were taken at the
midpoint of the central span plotted over the time.
Figure 5.4 - Dynamic influence lines of the vertical displacement at the midpoint of the central span.
70
Figure 5.5 - Dynamic influence lines of the vertical acceleration at the midpoint of the central span.
Considering the present case of analysis, it can be observed that the model which allows warping and
distortion has a maximum vertical displacement of 4.21 mm which compares to the value of 3.34 mm
of the model with the distortion retrained and to the 2.91 mm from the frame model. However, the
maximum acceleration is obtained in the model with the distortion restrained. In these analyses, as it
can be seen in Chapter 4, several factors, as span, velocity or the restraints applied to the cross section,
intervene in the results obtained.
The maximum values obtained in the example with a velocity of 200 kmh-1 will be part of a final
graphic representing an envelope of the maximum values achieved for each model according to the
speed.
5.4.2 HSLM-A10 - Box girder Analysis
A box girder is analysed through the deck models of a three-span bridge represented in Figure 5.3. Its
properties, plotted in Table 4.2, are considered in order to evaluate the dynamic response of the
structure. The bridge girder, Figure 5.2, is loaded with an eccentric high-speed train HSLM-A10
considering a range of velocities between 140 kmh-1and 420 kmh-1 and a damping coefficient of 1%.
71
The results of the vertical acceleration and vertical displacement of the box girder are illustrated in
Figures 5.6 to 5.9, which represent the maximum values at the midpoint of the central span according
to the velocity. The figures present the results for two different heights of the cross section, case h1
and case h2, whereas the analysis of the case h3 can be observed in Annex B1, since its peaks of the
velocity lie outside the range of velocities considered in this analysis.
Case 1- h1= 1.68m
Figure 5.6 - Envelope of maximum acceleration at midpoint of the central span according to the speed.
Figure 5.7 - Envelope of maximum displacement at midpoint of the central span according to the speed.
0
0.6
1.2
1.8
2.4
3
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Acc
ele
rati
on
[m
s-2]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
0
3
6
9
12
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Dis
pla
cem
en
t [m
m]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
72
Case 2- h2= 2.1m
Figure 5.8 - Envelope of maximum acceleration at midpoint of the central span according to the speed.
Figure 5.9 - Envelope of maximum displacement at midpoint of the central span according to the speed.
Comparing the results for the two box girders with different heights, it can be observed that the
maximums are obtained for different velocities. For the case h1, the peak of acceleration and
displacement occur at the velocity of 300kmh-1 whereas for the case h2 the peak occur at around
380kmh-1. Hence, it can be concluded that the higher the stiffness of the section, the higher the
velocity needs to be to have the maximum of acceleration and displacement.
0
0.6
1.2
1.8
2.4
3
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Acc
ele
rati
on
[m
s-2]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
0
3
6
9
12
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Dis
pla
cem
en
t [m
m]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
73
In relation to the local effects obtained by considering different modelling assumptions, it can be seen
in both cases that the distortion effect has an impact on the acceleration values. When this effect is
restrained, the peak of acceleration becomes higher comparing to the model where the distortion and
the warping are allowed, in which case, the section has more flexibility and shows a lower peak of
acceleration. With respect to the displacement, it can be observed that in both cases when the model
allowing distortion and warping the displacements are generally higher regardless the velocity.
However for the critical velocity, i.e. when the maximum displacement is achieved its magnitude
becomes closer to the shell model restraining the distortion and the frame model shows only a slight
difference. Thus, by comparison to the results obtained with the shell models, the frame model is
proved to be a reliable method of analysis providing accurate results.
5.4.3 HSLM-A10 - Double-T section Analysis
The study previously performed for a box girder is carried out, in this section, for a double-T cross
section with a much lower torsional stiffness for the purpose of comparing the results between these
two different cross sections.
The Double-T cross section properties are presented in Table 4.2, where the three different property
cases correspond to the three different section heights. The bridge girder is modelled by a three-span
beam, Figure 5.2, and it is loaded with an eccentric high-speed train HSLM-A10.
Figures 5.10 to 5.13 illustrate the maximum vertical accelerations and vertical displacements at the
mid-point of the central span for two different heights of the cross section, case h1 and case h2,
whereas the analysis of case h3 can be seen in Annex B2.
74
Case 1- h1= 2.53 m
Figure 5.10 - Envelope of maximum acceleration at midpoint of the central span according to the speed.
Figure 5.11 - Envelope of maximum displacement at midpoint of the central span according to the speed.
0
2.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Acc
ele
rati
on
[m
s-2]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
0
3
6
9
12
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Dis
pla
cem
en
t [m
m]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame Model
75
Case 2- h2= 3.03m
Figure 5.12 - Envelope of maximum displacement at midpoint of the central span according to the speed.
Figure 5.13 - Envelope of maximum displacement at midpoint of the central span according to the speed.
The dynamic analyses performed to the double-T section show that the maximum acceleration and
displacement does not occur for the same velocity in the different models considered. In the model
allowing warping and distortion, the one with the lowest natural frequency, the peak of acceleration
occurs for a lower velocity, whereas in the frame model, the stiffest and, therefore with the highest
natural frequency, the peak occurs for a higher velocity. The velocity producing a peak acceleration in
the shell model with the distortion constrained lies between the former two. Also, when the results of
0
2.4
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440
Acc
ele
rati
on
[m
s-2]
velocity [kmh-1]Shell model allowing warping/distortion Shell model restraining distortion Frame model
0
3
6
9
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440
Dis
pla
cem
en
t [m
m]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restrainig distortion Frame model
76
different cases of analysis, h1 and h2, are compared, it is evident that the case with the deeper cross
section experiences peak accelerations and deflections for higher velocities. Therefore, it can be
concluded that the stiffer the section is, the higher the velocity needs to be to generate a peak of
acceleration and displacement.
In regard to the local effects, constraining the distortion of the cross section impacts on the structural
behaviour shifting up the speed to which the maximum acceleration occurs but not reducing its
magnitude. However, it becomes effective in terms of controlling displacement as deflections appear
to be gradually lower as distortion is impeded.
In case 2, the critical speeds for the different models appear to come closer together. One can conclude
that as the cross section grows deeper, local effects begin to play less of a role in the dynamic response
of the bridge.
5.4.4 Concluding Remarks
The illustrative examples performed in this section have proven that the distortion make an effect on
the peak values obtained.
It can also be concluded that the higher is the stiffness of the section, the higher need to be the velocity
to have a peak of acceleration. This is noticed in both cross-section by observing the shift between the
analyses case 1and 2, and as consequence, the results for case 3 are presented just in annex since its
peaks of the velocity lie outside the range of velocities considered in this analysis.
To remark, the frame model performed results with good approximation compared to the shell models.
77
6. Concluding Remarks and Future Developments
The present dissertation presents a study on the dynamic response of railway bridges due to eccentric
high-speed trains, with the main objective to analyse the influence of the local effect of distortion in
the behaviour of the deck and on the maximum displacement and acceleration.
The results presented in this work, behind the analytical study, were provided by of finite elements
models which allowed to analyse and comprehend the behaviour of the cross section over the
longitudinal span, identifying the contribution of the warping and distortion in the behaviour of the
cross section.
The understanding of the dynamic behaviour began with the presentation of the formulation of
equations of motion, where an introduction of the natural frequencies and respective mode shapes was
done, and the methodologies used to evaluate the dynamic response of structures.
In what concerns to the forced vibrations due to moving vehicles, numerical and analytical models
were developed to obtain the results trough the coupled governing differential equations of moving
load obtained by Frýba, having been performed a modal analysing along with a direct integration
scheme.
There are several factors which influence the dynamic behaviour of the structure, as mass, stiffness,
damping, velocity, among others. A simply supported beam loaded by a moving load with different
velocities was analysed and consequently, the resonance effect was studied. It was concluded that
there is proporcionality between the stress resultant and the vertical displacements, since the variation
of the vertical displacement and bending moment along the span are qualitatively the same for any
considered velocity. However, the structure tends to show higher acceleration when higher velocities
are considered. The influence of different values of damping was also studied and it was possible to
conclude that this influence has a bigger impact when higher velocities are analysed. Those analytical
cases were compared with the numerical method of Newmark and with a finite element model. It was
78
confirmed a good accuracy between the displacements obtained through the different methods. In
relation to the acceleration, it was observed some differences, however, the maximum values of
acceleration obtained have the same trend and in all cases, the maximum value was the same.
Subsequently, a continuous beam was subjected to the moving loads with different velocities and
different cases of load having been observed the importance of the analyse of a set of loads instead of
a single load, in such a way that cause resonance and increased the structure answer.
Thereafter, a box cross section and a double-T cross section were analysed in detailed to understand
the importance of the distortion effect in the dynamic behaviour. By using the finite elements, shell
models allowing and restraining the distortional displacement were developed and a frame model was
also considered. Firstly, with the static analysis of an eccentric load, it was noticed that the effect of
the lower torsional stiffness of the opening section has significant influence in the way how these
sections deform. In this analysis, it was also observed the influence of the distortion since, considering
the shell model restraining the distortion there is an important difference when compared to the
displacement obtained through the model which allows the cross section to deform freely, these
differences are more visible when the box cross section is analysed. After that, a dynamic analysis due
to a single moving load showed, through an example, that the shell model allowing warping and
distortion gives a major value of vertical displacement. This was expected since this model has a
higher flexibility than the others, but the maximum value of acceleration is obtained at the shell model
which constrains the distortion.
Finally, a multi-span bridge subject to a high-speed train applied by different velocities were studied.
It was analysed the box cross section and the double-T section considering three different heights to
these cross sections which leads to different stiffness of the sections and, consequently, different
natural frequencies and dynamic responses. The results prove the importance of the distortion effect,
since the peak value of vertical acceleration and vertical displacement change, and when this effect is
neglected the vertical acceleration also increase. Hence, it can be concluded that the frame model is a
simple way to achieve results with goodapproximation, since the results obtained are in accordance
with the shell models.
79
Lastly, some suggestions aiming at future developments of the present work are proposed:
Consider another typology of cross-section and also analyse cases of steel-concrete
composite decks to compare the distortion effect;
Consider curved bridges in plane, with arbitrary types of cross-sections;
Consider the ballast track, to include the interaction of the track-structure;
Detail the support conditions defined at the real point of their applications, i.e. the
bearings of a real multi-span bridges at the pier sections;
Consider geometrical non-linear analysis of the deck.
80
7. References
[1] Amaral, Vasco M. (2009). Consideração dos efeitos dinâmicos no projecto de pontes em linhas
ferroviárias de altavelocidade, dissertação de Mestrado em Engenharia Estruturas. IST, Universidade
Técnica de Lisboa, Lisboa.
[2] Chan, T., & Ashebo, D. (2006). Theoretical study of moving force identification on continuous
bridges. Journal of Sound and Vibration, 295(3-5), 870-883.
[3] Chopra, A. (1995). Dynamics of Structures - Theory and application to the Earthquake
Engineering. Berkeley: Prentice Hall, Inc.
[4] Clough, R.W. and Penzien, J. (1995). Dynamics of Structures, Computers and Structures, Inc.,
Berkeley, USA.
[5] EN 1990-A2 (2005): Basis of structural Design - Annex A2: Applications for bridges, CEN –
European Committee for Standardization.
[6] EN 1991-2 (2003): Actions on structures – Part 2: Traffic loads on bridges, CEN – European
Committee for Standardization.
[7] ERRI D214 (1999). Rail Bridges for Speed > 200 km/h, Final Report, Part A, Synthesis of The
Result of D 214 research, Part B, Proposed UIC Leaflet, European Rail Research Institute ERRI.
[8] Frýba, L. (1972). Vibration of Solids and Structures under Moving Loads. Thomas Telford.
[9] Frýba, L. (1996). Dynamics of Railway Bridges, Thomas Telford Services Ltd., Prague.
[10] Frýba, L. (2001) – "A rough assessment of railway bridges for high-speed trains", Engineering
Structures nº23.
[11] Graça, A. (2011). Modelação da Resposta em Pontes Devido à Passagem de Comboios de Alta
Velocidade – formulação e implementação computacional de um modelo para simulação do efeito da
81
passagem de comboios de altavelocidade em pontes, Master Dissertation, Instituto Superior Técnico,
Lisboa.
[12] Ichikawa, M., Matsuda, A. and Miyakawa, Y. (1999). Simple analysis of a multi-span beam
under moving loads with variable velocity, Transactions of the Japan Society for Aeronautical and
Spaces Sciences, 41, 168-173.
[13] Ichikawa, M., Miyakawa, Y. and Matsuda, A. (2000). Vibration analysis of the continuous beam
subjected to a moving mass, Journal of Sound and Vibration, 230(3), 493-506.
[14] Lee, H.P. (1996(1)). Dynamic response of a beam with a moving mass, Journal of Sound and
Vibration, 191, 289-294.
[15] Lee, H.P. (1996(2)). Transverse vibration of a Timoshenko beam acted upon by an accelerating
mass, Applied Acoustics, 47, 319-330.
[16] Lee, H.P. (1996(3)). Dynamic response of a beam on multiple supports with a moving mass,
Journal of Structural Engineering and Mechanics, 4, 303-312.
[17] Lisi, Diego (2011). A Beam Finite Element Model Including Warping – Application to the
Dynamic and Static Analysis of Bridge Decks, Instituto Superior Técnico, Lisboa.
[18] Michaltsos, G.T., Sophianopoulos, D. and Kounadis, A.N. (1996). The effect of a moving mass
and other parameters on the dynamic response of a simply supported beam, Journal of Sound and
Vibration, 191, 357-362.
[19] Olsson, M. (1991). On the fundamental moving mass problem, Journal of Sound and Vibration,
145, 299-307.
[20] Ouyang, H. (2011). Moving-load dynamic problems: A tutorial (with a brief overview),
Mechanical Systems and Signal Processing, 25, 2039-2060.
[21] Serra, J. (2014). Dynamic Analysis of Bridge Girders Subjected to Moving Loads-Numerical and
Analytical Beam Models Considering Warping Effects, Master Dissertation, Instituto Superior
Técnico, Lisboa.
82
[22] Timoshenko, S. P. (1922). On forced vibration of bridges. Phil. Mag., 6, 1018-1019.
[23] Yang, Y., & Lin, C. (2004). Vehicle bridge interaction dynamics with applications to high-speed
railways.(W. Scientific, Ed.)Journal of Sound and Vibration (Vol. 284).
[24] Yang, Y.B., Yau, J.D. and Wu, Y.S. (2004). Vehicle-Bridge Interaction Dynamics, World
Scientific Publishing Co.
[25] Yang, Y.B., Liao, S.S. and Lin, B.H. (1995). Impact formulas for vehicles moving over simple
and continuous beams, Journal of Structural Engineering, American Society of Civil Engineers, 121,
1644-1650.
[26] Willis, R. (1849). Appendix to the report of the commissioners appointed to inquire into the
application of iron to railway structures. Stationery Office, London.
[27] Zhu, X.Q. and Law, S.S. (2001). Precise time-step integration for the dynamic response of a
continuous beam under moving loads, Journal of Sound and Vibration, 240, 962-970.
83
84
8. Annex
A1 - VBA Code
Sub SortALLsheets()
DimwsheetAs Worksheet
For Each wsheet In ActiveWorkbook.Worksheets
Sheets(wsheet.Name).Select
Sheets(wsheet.Name).Range("K10:K1000").Clear
Dim t As Double
Dim P As Double
Dim Q As Double
Dim tempo As Double
Dim sum As Double
DimlinAs Integer
Dim lin2 As Integer
Dim lin3 As Integer
lin = 9
lin2 = 12
lin3 = 9
t = 0
tempo = Cells(lin2, 3)
sum = 0
While Cells(lin2, 3).Value <> ""
While t <= 1
Q = Cells(lin, 7)
Cells(lin3, 11) = Q + Cells(lin3, 11)
lin = lin + 1
lin3 = lin3 + 1
t = t + 0.002
Wend
t = 0
lin = 9
lin2 = lin2 + 1
tempo = Cells(lin2, 3)
85
lin3 = 9 + tempo / 0.002
Wend
Next wsheet
End Sub
A2 - VBA Code
Sub Macro1()
Dim tAs Double
Dim PAs Double
Dim QAs Double
Dim tempo As Double
Dim sum As Double
Dim linAs Integer
Dim lin2 As Integer
Dim lin3 As Integer
lin = 9
lin2 = 12
lin3 = 9
t = 0
tempo = Cells(lin2, 3)
sum = 0
While Cells(lin2, 3).Value <> ""
While t<= 1.286
Q = Cells(lin, 7)
Cells(lin3, 11) = Q + Cells(lin3, 11)
lin = lin + 1
lin3 = lin3 + 1
t = t + 0.002
Wend
t = 0
lin = 9
lin2 = lin2 + 1
tempo = Cells(lin2, 3)
lin3 = 9 + tempo / 0.002
Wend
End Sub
86
B1 - HSLM-A10 Box Girder Section
Case h3- h=2.68m
Figure B1.1. - Envelope of maximum acceleration at midpoint of the central span according to the speed.
Figure B1.2. - Envelope of maximum displacement at midpoint of the central span according to the speed.
0
3
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Acc
ele
rati
on
[m
s-2]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
0
3
6
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Dis
pla
cem
en
t [m
m]
velocity [kmh-1]Shell model allowing warping/distortion Shell model restraining distortion Frame model
87
B2 - HSLM-A10 Double-T Section
Case h3- h=3.03m
Figure B2.1. - Envelope of maximum acceleration at midpoint of the central span according to the speed.
Figure B2.2 - Envelope of maximum displacement at midpoint of the central span according to the speed.
0
3
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Acc
ele
rati
on
[m
s-2
]
velocity [kmh-1]
Shell model allowing warping/distortion Shell model restraining distortion Frame model
0
3
140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
Dis
pla
cem
en
t [m
m]
velocity [km-1]Shell model allowing warping/distortion Shell model restraining distortion Frame model